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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

sweety439 2021-01-07 14:32

some 2-digits number xy in base b such that the smallest prime of the form xy*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that both x*b^n+1 and y*b^n+1 are ruled out to only contain composites)

b=11:

[CODE]
12: 1453
16: 3429742097
18: 199
20: 56204873696128495701224278067068097021
36: 397
38: 419
40: 8574355241
42: 463
48: 6922076933111963233808451075889
56: 617
58: 7019
60: 661
62: 683
78: 859
80: 881
82: 9923
86: 947
100: 12101
102: 1123
106: 187785467
108: 191328589
[/CODE]

b=13:

[CODE]
16: 35153
18: 32258887092667
22: 3719
24: 313
28: 4733
34: 443
40: 521
42: 547
46: 599
48: 48*13^6267+1
66: 859
70: 911
72: 937
76: 166973
94: 1223
96: 1249
100: 1301
102: 1327
106: 25476138101289768358558528047514833801370032844988650960096232747
112: 2609385533717873
118: 211474926496367
120: 120*13^1552+1
124: 1613
126: 276823
144: 1873
148: 25013
150: 1951
154: 2003
[/CODE]

sweety439 2021-01-07 15:13

some 3-digits number xyz in base b such that the smallest prime of the form xyz*b^n+1 is minimal prime (start with b+1) in base b: (they have the property that all of x*b^n+1, y*b^n+1, z*b^n+1, xy*b^n+1, xz*b^n+1, yz*b^n+1 are ruled out to only contain composites)

base 11:

[CODE]
158: 210299
168: 27056569
378: 4159
408: 49369
498: 5479
518: 689459
548: 6029
568: 68729
588: 6469
658: 1165687139
708: 7789
898: 103832913046615294450266950864953559038103755928866797157252835645367668431847219
928: 112289
1138: 137699
1148: 124382424230098512904021128149
[/CODE]

base 13:

[CODE]
184: 2393
190: 917093711
196: 2549
202: 75001187
262: 3407
268: 45293
274: 46307
280: 615161
352: 10053473
358: 292031598119
382: 4967
388: 852437
406: 5279
412: 152972717
430: 5591
436: 5669
460: 5981
466: 13309427
[no known prime for S13 k=484, 2B3{0}1 is unsolved family in base 13]
490: 82811
538: 1181987
544: 1195169
574: 97007
580: 7541
616: 8009
622: 8087
652: 531856430093
658: 90710887636643
874: 264712843161629123
880: 148721
886: 11519
892: 11597
952: 12377
958: 2104727
964: 162917
970: 12611
1198: 34216079
1204: 2645189
1210: 15731
1216: 15809
1276: 80067107693
1282: 36615203
1288: 2829737
1294: 16823
1366: 652758153339229918146262170195611899412691250234661073608281535607441327245782154274226163627511856910242680997040239850166408242104102157590110059393064877741749052590786638302327896695517321882076385248095689607714279245947320040491619053213124101066946141643999231501774298529691499984819614064572813909251081765622950846989229615735750095118621279499641063473790506905622553548920945551313745769942678628213020977681299198153663054881959698284804196609230966508441216040759006656791607
1372: 17837
1396: 18149
1402: 12127883118972635782067
1420: 18461
1426: 18539
1444: 18773
1450: 41413451
1474: 19163
1480: 448255157756534441
1498: 34900531513476539
1504: 19553
1552: 20177
1558: 263303
1588: 2321529421116208423755077
1594: 269387
1630: 21191
1636: 21269
1666: 47582627
1672: 21737
1888: 821828298004735653739505427655167983139612951149032024600411489000613133621406169905393832026983244445467247164419724522787738176502646863025015739869100207704101192365412789485747731430993843624803368161051869616583490868130229284719241207489969244826144259775702398961583808046895149627729346180500126934284621986296138708001440861667299169
1894: 24623
1900: 1549888369901
1906: 4187483
1966: 4319303
1972: 333269
1978: 56493659
1984: 25793
[/CODE]

sweety439 2021-01-07 15:30

[QUOTE=sweety439;531436][URL="https://primes.utm.edu/glossary/page.php?sort=MinimalPrime"]https://primes.utm.edu/glossary/page.php?sort=MinimalPrime[/URL]

In 1996, Jeffrey Shallit [Shallit96] suggested that we view prime numbers as strings of digits. He then used concepts from formal language theory to define an interesting set of primes called the minimal primes:

For example, if our set is the set of prime numbers (written in radix 10), then we get the set {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, and if our set is the set of composite numbers (written in radix 10), then we get the set {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}

Besides, if our set is the set of prime numbers written in radix b, then we get these sets:

Now, let's consider: if our set is [B]the set of prime numbers >= b[/B] written in radix b (i.e. the prime numbers with at least two digits in radix b), then we get the sets:
[/QUOTE]

This puzzle is an extension of the original [URL="https://www.primepuzzles.net/puzzles/puzz_178.htm"]minimal prime base b puzzle[/URL], to include CRUS [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski[/URL]/[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel[/URL] conjectures base b with k-values < b

The original minimal prime base b puzzle does not cover CRUS Sierpinski/Riesel conjectures base b with CK < b (such Riesel bases are 14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 77, 81, 83, 84, 86, 89, 90, 92, 94, 98, 104, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 155, 158, 164, 167, 170, 173, 174, 176, 178, 179, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 284, 285, 286, 289, 290, 293, 294, 296, 298, 299, 300, 302, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 371, 373, 374, 376, 377, 379, 380, 383, 384, 386, 387, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 407, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 470, 472, 473, 474, 475, 476, 479, 480, 482, 484, 488, 489, 491, 492, 493, 494, 497, 500, 503, 504, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 523, 524, 527, 528, 529, 530, 531, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 577, 578, 579, 580, 581, 582, 584, 587, 588, 590, 593, 594, 596, 597, 599, 602, 604, 605, 608, 609, 611, 614, 615, 617, 619, 620, 622, 623, 626, 628, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 668, 669, 670, 671, 674, 676, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 694, 695, 696, 698, 699, 701, 702, 704, 706, 707, 710, 712, 713, 714, 716, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 737, 739, 740, 741, 743, 744, 746, 747, 749, 752, 753, 755, 758, 759, 761, 762, 764, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 961, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 988, 989, 992, 993, 994, 995, 998, 1000, 1002, 1003, 1004, 1007, 1010, 1011, 1013, 1014, 1016, 1017, 1019, 1021, 1022, 1024, ..., and such Sierpinski bases are 14, 20, 29, 32, 34, 38, 41, 44, 47, 50, 54, 56, 59, 62, 64, 65, 68, 69, 74, 76, 77, 83, 84, 86, 89, 90, 92, 94, 98, 101, 104, 109, 110, 113, 114, 116, 118, 119, 122, 125, 128, 129, 131, 132, 134, 137, 139, 140, 142, 144, 146, 149, 152, 153, 154, 155, 158, 159, 160, 164, 167, 169, 170, 172, 173, 174, 176, 179, 181, 182, 184, 185, 186, 188, 189, 194, 197, 200, 202, 203, 204, 206, 208, 209, 212, 214, 216, 218, 219, 220, 221, 224, 227, 229, 230, 233, 234, 236, 237, 239, 242, 244, 245, 246, 248, 251, 252, 254, 257, 258, 259, 260, 263, 264, 265, 266, 269, 272, 274, 275, 278, 279, 281, 284, 285, 289, 290, 293, 294, 296, 298, 299, 300, 302, 304, 305, 307, 308, 309, 311, 314, 317, 318, 320, 321, 322, 324, 326, 328, 329, 332, 334, 335, 338, 339, 340, 341, 344, 347, 349, 350, 353, 354, 356, 359, 362, 363, 364, 365, 368, 369, 370, 371, 373, 374, 377, 379, 380, 384, 386, 389, 390, 392, 394, 395, 398, 401, 402, 404, 405, 406, 407, 409, 410, 412, 413, 414, 416, 417, 419, 422, 424, 425, 426, 428, 429, 433, 434, 436, 437, 439, 440, 441, 443, 444, 446, 447, 449, 450, 452, 454, 455, 458, 459, 461, 464, 467, 468, 469, 470, 472, 473, 474, 475, 476, 479, 480, 482, 483, 484, 488, 489, 491, 492, 493, 494, 496, 497, 500, 501, 503, 504, 505, 506, 509, 510, 512, 514, 515, 516, 517, 518, 519, 521, 524, 526, 527, 528, 530, 531, 532, 533, 534, 536, 538, 539, 542, 544, 545, 548, 549, 550, 551, 552, 554, 557, 558, 559, 560, 562, 563, 564, 566, 569, 571, 572, 573, 574, 578, 579, 580, 581, 582, 584, 587, 588, 589, 590, 593, 594, 596, 597, 599, 601, 602, 604, 605, 608, 609, 610, 611, 614, 615, 617, 619, 620, 622, 623, 626, 629, 632, 634, 635, 636, 637, 638, 641, 643, 644, 645, 647, 648, 649, 650, 653, 654, 656, 657, 659, 662, 664, 665, 666, 668, 669, 670, 671, 674, 677, 678, 679, 680, 681, 683, 684, 686, 688, 689, 692, 695, 696, 698, 699, 701, 702, 703, 704, 706, 707, 709, 710, 712, 713, 714, 716, 718, 719, 720, 722, 724, 725, 727, 729, 730, 731, 734, 736, 737, 739, 740, 741, 743, 744, 746, 747, 748, 749, 752, 753, 754, 755, 758, 759, 761, 762, 764, 766, 767, 769, 770, 773, 774, 776, 778, 779, 780, 781, 782, 783, 784, 785, 788, 789, 790, 791, 792, 794, 797, 798, 800, 802, 803, 804, 805, 806, 809, 811, 812, 813, 814, 815, 816, 818, 819, 821, 824, 825, 827, 828, 829, 830, 832, 833, 834, 835, 836, 839, 842, 844, 845, 846, 848, 849, 850, 851, 853, 854, 857, 859, 860, 863, 864, 866, 867, 868, 869, 870, 872, 873, 874, 875, 878, 879, 881, 883, 884, 887, 888, 889, 890, 892, 893, 894, 896, 898, 899, 900, 901, 902, 903, 904, 905, 908, 909, 911, 912, 914, 916, 917, 919, 920, 922, 923, 924, 926, 929, 930, 932, 934, 935, 937, 938, 939, 941, 942, 944, 945, 947, 948, 949, 950, 951, 953, 954, 956, 958, 959, 962, 964, 965, 967, 968, 969, 972, 974, 977, 978, 979, 980, 983, 984, 985, 986, 987, 989, 992, 993, 994, 995, 998, 1000, 1001, 1004, 1006, 1007, 1009, 1010, 1011, 1013, 1014, 1016, 1019, 1022, 1024, ...), since in Riesel side, the prime is not minimal prime if either k-1 or b-1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime if k is prime (e.g. 25*30^34205-1 is not minimal prime in base 30, since it is OT[SUB]34205[/SUB] in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if single-digit primes are not counted), but this extended version of minimal prime base b problem does, this requires a restriction of prime >= b, i.e. primes should have >=2 digits, and the single-digit primes (including the k-1, b-1, k) are not allowed.

sweety439 2021-01-07 15:34

CRUS requires exponent n>=1 for these primes, n=0 is not acceptable to avoid the trivial primes (e.g. 2*b^n+1, 4*b^n+1, 6*b^n+1, 10*b^n+1, 12*b^n+1, 16*b^n+1, 3*b^n-1, 4*b^n-1, 6*b^n-1, 8*b^n-1, 12*b^n-1, 14*b^n-1, ... cannot be quickly eliminated with n=0, or the conjectures become much more easy and uninteresting)

For the same reason, this minimal prime puzzle requires >=base (i.e. >=2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. simple families containing digit 2, 3, 5, 7, B, D, H, J, N, T, V, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much more easy and uninteresting)

sweety439 2021-01-07 19:19

[QUOTE=sweety439;567582]Base b minimal primes (start with 2 digits) includes: [/QUOTE]

Some known minimal primes (start with b+1) and unsolved families for large bases b:

* For the repunit case (family {(1)}), see [URL="https://mersenneforum.org/attachment.php?attachmentid=23101&d=1597771406"]https://mersenneforum.org/attachment.php?attachmentid=23101&d=1597771406[/URL] and [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]

* Unsolved family {(1)} in bases b = 185, 269, 281, 380, 384, 385, 394, 396, 452, 465, 511, 574, 598, 601, 629, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (less than 1024)

* Unsolved family (40):{(121)} in base 243

* For the GFN case (family (1){(0)}(1)), see [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL] and [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]

* Unsolved family (1){(0)}(1) in bases b = 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (less than 1024)

* Unsolved families (4){(0)}(1) and (16){(0)}(1) in base 32, (16){(0)}(1) in base 128, (36){(0)}(1) in base 216, (2){(0)}(1), (4){(0)}(1), (16){(0)}(1), (32){(0)}(1), (256){(0)}(1) in base 512, (10){(0)}(1) and (100){(0)}(1) in base 1000, (4){(0)}(1) and (16){(0)}(1) in base 1024

* Unsolved families {((b-1)/2)}((b+1)/2) in base b = 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (less than 1024)

* Unsolved family (12):{(62)}:(63) in base 125, (24):{(171)}:(172) in base 343

* For the Williams 1st case (family (b-2){(b-1)}), see [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL] and [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL]

* Unsolved family (b-2){(b-1)} in bases b = 128, 233, 268, 293, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (less than 1024)

* For the Williams 2nd case (family (b-1){(0)}1), see [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL]

* Unsolved family (b-1){(0)}1 in bases b = 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (less than 1024)

* For the Williams 4th case (family (1)(1){(0)}(1)), see [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]

* Unsolved family (1)(1){(0)}(1) in bases 813, 863, 1017 (not base 962, since in base 962, (1)(0)(0)(0)(1) is prime) (less than 1024)

* Minimal primes (70)[SUB]3018[/SUB](1) in base 71, (81)[SUB]168[/SUB](1) in base 82, (82)[SUB]964[/SUB](1) in base 83, (87)[SUB]2847[/SUB](1) in base 88, (113)[SUB]990[/SUB](1) in base 114, (127)[SUB]400[/SUB](1) in base 128, (142)[SUB]281[/SUB](1) in base 143, (144)[SUB]254[/SUB](1) in base 145

* Unsolved family {(92)}(1) in base 93 and {(112)}(1) in base 113, {(151)}(1) in base 152, {(157)}(1) in base 158

* Minimal primes (1)(0)[SUB]193[/SUB](79) in base 80, (1)(0)[SUB]1399[/SUB](106) in base 107, (1)(0)[SUB]20087[/SUB](112) in base 113, (1)(0)[SUB]64369[/SUB](122) in base 123, (1)(0)[SUB]503[/SUB](127) in base 128, (1)(0)[SUB]103[/SUB](160) in base 161

* For the (2){(0)}(1), (3){(0)}(1), (4){(0)}(1), ..., (12){(0)}(1) case, see [URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL]

* Unsolved family (2){(0)}(1) in bases 365, 383, 461, 512 (GFN), 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004

* Unsolved family (3){(0)}(1) in bases 718, 912

etc.

* For the (1){(b-1)}, (2){(b-1)}, (3){(b-1)}, ..., (11){(b-1)} case, see [URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL]

* Unsolved family (1){(580)} in base 581, (1){(991)} in base 992, (1){(1018)} in base 1019

* Unsolved family (2){(587)} in base 588, (2){(971)} in base 972

etc.

* Minimal primes (1)(0)[SUB]112[/SUB](2) in base 47, (1)(0)[SUB]254[/SUB](2) in base 89, (1)(0)[SUB]135[/SUB](2) in base 159

* Unsolved family (1){(0)}(2) in base 167

* Minimal primes (80)[SUB]129[/SUB](79) in base 81, (96)[SUB]746[/SUB](95) in base 97, (196)[SUB]163[/SUB](195) in base 197, (208)[SUB]125[/SUB](207) in base 209, (214)[SUB]133[/SUB](213) in base 215, (220)[SUB]551[/SUB](219) in base 221, (286)[SUB]3409[/SUB](285) in base 287

* Unsolved family {(304)}(303) in base 305

* For k*b^n+1, see [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL], all 1<=k<=b-1 are minimal primes or unsolved families

** Also, all two-digit (when written in base b) k-values while both digits d of k cannot have prime of the form d*b^n+1 are minimal primes or unsolved families

* For k*b^n-1, see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL], all 1<=k<=b-1 are minimal primes or unsolved families

** Also, all two-digit (when written in base b) k-values while both digits d of k-1 cannot have prime of the form (d+1)*b^n-1 are minimal primes or unsolved families

sweety439 2021-01-07 19:32

Examples of families which can be proven to contain no primes > base (and no subsequences of these families can be primes > base):

(using 0-9 for digit values 0-9, A-Z for digit values 10-35, a-z for digit values 36-61, "+" for digit value 62, "/" for digit value 63, for bases <=64; and using decimal to represent individual digits and using ":" as separating mark, for bases >64)

* all families with ending number not coprime to base (all such numbers are not coprime to base, thus are either composite or factors of base (thus <= base), thus cannot be primes > base)
* all families with gcd of the digits > 1 (all such numbers are divisible by the gcd, and since the gcd is < base, all such numbers are either composite or equal to the gcd (thus also < base), thus cannot be primes > base)
* 1{0}1 in base 3 (all such numbers are divisible by 2)
* 2{0}1 in base 4 (all such numbers are divisible by 3)
* 1{0}1 in base 5 (all such numbers are divisible by 2)
* 1{0}3 in base 5 (all such numbers are divisible by 2)
* 3{0}1 in base 5 (all such numbers are divisible by 2)
* 4{0}1 in base 6 (all such numbers are divisible by 5)
* 1{0}1 in base 8 (all such numbers factored as sum of cubes)
* {1} in base 9 (all such numbers factored as difference of squares)
* {1}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* 2{7} in base 9 (all such numbers are divisible either by 2 or by 5)
* 3{1} in base 9 (all such numbers factored as difference of squares)
* {3}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* {3}8 in base 9 (all such numbers are divisible either by 2 or by 5)
* 3{8} in base 9 (all such numbers factored as difference of squares)
* 5{1} in base 9 (all such numbers are divisible either by 2 or by 5)
* 5{7} in base 9 (all such numbers are divisible either by 2 or by 5)
* 6{1} in base 9 (all such numbers are divisible either by 2 or by 5)
* {7}2 in base 9 (all such numbers are divisible either by 2 or by 5)
* {7}5 in base 9 (all such numbers are divisible either by 2 or by 5)
* 8{3} in base 9 (all such numbers are divisible either by 2 or by 5)
* {8}5 in base 9 (all such numbers factored as difference of squares)
* 4{6}9 in base 10 (all such numbers are divisible by 7)
* 2{5} in base 11 (all such numbers are divisible either by 2 or by 3)
* 3{5} in base 11 (all such numbers are divisible either by 2 or by 3)
* 3{7} in base 11 (all such numbers are divisible either by 2 or by 3)
* 4{7} in base 11 (all such numbers are divisible either by 2 or by 3)
* {B}9B in base 12 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 13)
* 1{0}B in base 14 (all such numbers are divisible either by 3 or by 5)
* 3{D} in base 14 (all such numbers are divisible either by 3 or by 5)
* 4{0}1 in base 14 (all such numbers are divisible either by 3 or by 5)
* 8{D} (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* A{D} in base 14 (all such numbers are divisible either by 3 or by 5)
* B{0}1 in base 14 (all such numbers are divisible either by 3 or by 5)
* {D}3 in base 14 (all such numbers are divisible either by 3 or by 5)
* {D}5 in base 14 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 9{6}8 in base 15 (all such numbers are divisible by 11)
* 1{5} in base 16 (all such numbers factored as difference of squares)
* {4}1 in base 16 (all such numbers factored as difference of squares)
* {4}D in base 16 (all such numbers are divisible by 3, 7, or 13)
* {8}F in base 16 (all such numbers are divisible by 3, 7, or 13)
* 8{F} in base 16 (all such numbers factored as difference of squares)
* B{4}1 in base 16 (all such numbers factored as difference of squares)
* {C}B in base 16 (all such numbers factored as difference of squares)
* {C}D in base 16 (all such numbers factored as x^4+4*y^4)
* {C}DD in base 16 (all such numbers factored as x^4+4*y^4)
* 1{9} in base 17 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 1{6} in base 19 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 1{0}D in base 20 (all such numbers are divisible either by 3 or by 7)
* 7{J} in base 20 (all such numbers are divisible either by 3 or by 7)
* 8{0}1 in base 20 (all such numbers are divisible either by 3 or by 7)
* C{J} in base 20 (all such numbers are divisible either by 3 or by 7)
* D{0}1 in base 20 (all such numbers are divisible either by 3 or by 7)
* {J}7 in base 20 (all such numbers are divisible either by 3 or by 7)
* 3{N} in base 24 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 5{N} in base 24 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 8{N} in base 24 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* {1} in base 25 (all such numbers factored as difference of squares)
* 2{1} in base 25 (all such numbers factored as difference of squares)
* 1{3} in base 25 (all such numbers factored as difference of squares)
* 1{8} in base 25 (all such numbers factored as difference of squares)
* 5{1} in base 25 (all such numbers factored as difference of squares)
* 5{8} in base 25 (all such numbers factored as difference of squares)
* 7{1} in base 25 (all such numbers factored as difference of squares)
* A{3} in base 25 (all such numbers factored as difference of squares)
* L{8} in base 25 (all such numbers factored as difference of squares)
* 1{0}8 in base 27 (all such numbers factored as sum of cubes)
* 7{Q} in base 27 (all such numbers factored as difference of cubes)
* 8{0}1 in base 27 (all such numbers factored as sum of cubes)
* 9{G} in base 27 (all such numbers factored as sum of cubes)
* {D}E in base 27 (all such numbers factored as sum of cubes)
* {Q}J in base 27 (all such numbers factored as difference of cubes)
* 1{0}1 in base 32 (all such numbers factored as sum of 5th powers)
* {1} in base 32 (all such numbers factored as difference of 5th powers)
* F{W} in base 33 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 17)
* 1{B} in base 34 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 8{X} in base 34 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* {X}P in base 34 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 3{7} in base 36 (all such numbers factored as difference of squares)
* 3{Z} in base 36 (all such numbers factored as difference of squares)
* 8{Z} in base 36 (all such numbers factored as difference of squares)
* O{Z} in base 36 (all such numbers factored as difference of squares)
* {Z}B in base 36 (all such numbers factored as difference of squares)
* C{b} in base 38 (all such numbers are divisible by 3, 5, or 17)
* G{0}1 in base 38 (all such numbers are divisible by 3, 5, or 17)
* 3{c} in base 39 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 1{9} in base 41 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 1{0}8 in base 47 (all such numbers are divisible by 3, 5, or 13)
* 1{0}G in base 47 (all such numbers are divisible by 3, 5, or 17)
* 8{0}1 in base 47 (all such numbers are divisible by 3, 5, or 13)
* D{k} in base 47 (all such numbers are divisible by 3, 5, or 13)
* G{0}1 in base 47 (all such numbers are divisible by 3, 5, or 17)
* H{n} in base 50 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 17)
* 3{r} in base 54 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* 5{r} in base 54 (all such numbers with even length factored as difference of squares, and all such numbers with odd length are divisible by 5)
* 8{r} in base 54 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 5)
* jP{0}1 in base 55 (all such numbers with length == 2 mod 4 factored as x^4+4*y^4, all such numbers with odd length are divisible by 7, and all such numbers with length == 0 mod 4 are divisible by 17)
* 1{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* 3{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* F{7} in base 57 (all such numbers with odd length factored as difference of squares, and all such numbers with even length are divisible by 2)
* (12):{(64)}:(65) in base 81 (all such numbers factored as x^4+4*y^4)
* {(64)}:(65) in base 81 (all such numbers factored as x^4+4*y^4)
* (73):{(80)} in base 81 (all such numbers are divisible by 7, 13, or 73)
* (8):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-4, 2))+1, and 7*n-4 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (32):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-2, 2))+1, and 7*n-2 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (64):{(0)}:(1) in base 128 (such numbers with length n are divisible by 2^(2^valuation(7*n-1, 2))+1, and 7*n-1 cannot be power of 2, since all powers of 2 are == 1, 2, 4 mod 7)
* (16):{(0)}:(1) in base 200 (all such numbers with length == 3 mod 4 factored as x^4+4*y^4, all such numbers with even length are divisible by 7, and all such numbers with length == 1 mod 4 are divisible by 17)
* (73):{(337)} in base 338 (all such numbers are divisible by 3, 5, or 73)
* (21):{(130)} in base 391 (all such numbers with odd length factored as difference of squares, all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 3 are divisible by 19, and all such numbers with length == 0 mod 6 are divisible by 109)
* (73):{(391)} in base 392 (all such numbers are divisible by 3, 5, or 73)
* (73):{(445)} in base 446 (all such numbers are divisible by 3, 7, 13, or 73)
* (1):(399):{(0)}:(1) in base 625 (all such numbers factored as x^4+4*y^4)
* (4):{(0)}:(1) in base 625 (all such numbers factored as x^4+4*y^4)
* (63):{(935)} in base 936 (all such numbers with odd length factored as difference of squares, all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 6 are divisible by 37, and all such numbers with length == 0 mod 6 are divisible by 109)
* (63):{(956)} in base 957 (all such numbers with length == 1 mod 3 factored as difference of cubes, all such numbers with length == 2 mod 3 are divisible by 73, and all such numbers with length == 0 mod 3 are divisible by 19)

sweety439 2021-01-08 10:57

1 Attachment(s)
Update pdf file for the proofs (not complete, continue updating ....)

sweety439 2021-01-08 17:02

Now, I try to prove base 12 (may find some minimal primes not in my current list)

In base 12, the possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are

(1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B)

* Case (1,1):

** [B]11[/B] is prime, and thus the only minimal prime in this family.

* Case (1,5):

** [B]15[/B] is prime, and thus the only minimal prime in this family.

* Case (1,7):

** [B]17[/B] is prime, and thus the only minimal prime in this family.

* Case (1,B):

** [B]1B[/B] is prime, and thus the only minimal prime in this family.

* Case (2,1):

** Since 25, 27, 11, 31, 51, 61, 81, 91, [B]221[/B], [B]241[/B], [B]2A1[/B], [B]2B1[/B] are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes)

*** The smallest prime of the form 2{0}1 is [B]2001[/B]

* Case (2,5):

** [B]25[/B] is prime, and thus the only minimal prime in this family.

* Case (2,7):

** [B]27[/B] is prime, and thus the only minimal prime in this family.

* Case (2,B):

** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, [B]2BB[/B] are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)

*** Since 90B, [B]200B[/B], [B]202B[/B], [B]222B[/B], [B]229B[/B], [B]292B[/B], [B]299B[/B] are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes)

**** None of 20B, 22B, 29B, 209B, 220B are primes.

* Case (3,1):

** [B]31[/B] is prime, and thus the only minimal prime in this family.

* Case (3,5):

** [B]35[/B] is prime, and thus the only minimal prime in this family.

* Case (3,7):

** [B]37[/B] is prime, and thus the only minimal prime in this family.

* Case (3,B):

** [B]3B[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-01-08 17:11

* Case (4,1):

** Since 45, 4B, 11, 31, 51, 61, 81, 91, [B]401[/B], [B]421[/B], [B]471[/B] are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since A41 and [B]4441[/B] are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes)

**** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form 44{A}1 is [B]44AAA1[/B]

* Case (4,5):

** [B]45[/B] is prime, and thus the only minimal prime in this family.

* Case (4,7):

** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, [B]447[/B], [B]497[/B] are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)

*** Since [B]4707[/B] and [B]4777[/B] are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes)

**** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime.

**** The smallest prime of the form 4{0}77 is [B]400000000000000000000000000000000000000077[/B]

* Case (4,B):

** [B]4B[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-01-08 17:25

* Case (5,1):

** [B]51[/B] is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, [B]565[/B] are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes)

*** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime.

* Case (5,7):

** [B]57[/B] is prime, and thus the only minimal prime in this family.

* Case (5,B):

** [B]5B[/B] is prime, and thus the only minimal prime in this family.

* Case (6,1):

** [B]61[/B] is prime, and thus the only minimal prime in this family.

* Case (6,5):

** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, [B]655[/B], [B]665[/B] are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since [B]6A05[/B] and [B]6AA5[/B] are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes)

**** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime.

**** The smallest prime of the form 6{0}A5 is [B]600A5[/B]

* Case (6,7):

** [B]67[/B] is prime, and thus the only minimal prime in this family.

* Case (6,B):

** [B]6B[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-01-08 17:54

* Case (7,1):

** Since 75, 11, 31, 51, 61, 81, 91, [B]701[/B], [B]721[/B], [B]771[/B], [B]7A1[/B] are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes)

*** Since 7BB, [B]7441[/B] and [B]7B41[/B] are primes, we only need to consider the numbers 741, 7B1, 74B1

**** None of 741, 7B1, 74B1 are primes.

* Case (7,5):

** [B]75[/B] is prime, and thus the only minimal prime in this family.

* Case (7,7):

** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, [B]747[/B], [B]797[/B] are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes)

*** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime.

* Case (7,B):

** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, [B]70B[/B], [B]77B[/B], [B]7BB[/B] are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes)

*** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes)

**** The smallest prime of the form 7{9}B is [B]7999B[/B]

**** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes)

**** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime)

* Case (8,1):

** [B]81[/B] is prime, and thus the only minimal prime in this family.

* Case (8,5):

** [B]85[/B] is prime, and thus the only minimal prime in this family.

* Case (8,7):

** [B]87[/B] is prime, and thus the only minimal prime in this family.

* Case (8,B):

** [B]8B[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-01-08 19:43

* Case (9,1):

** [B]91[/B] is prime, and thus the only minimal prime in this family.

* Case (9,5):

** [B]95[/B] is prime, and thus the only minimal prime in this family.

* Case (9,7):

** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, [B]907[/B] are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes)

*** Since 447, 497, 747, 797, [B]9777[/B], [B]9947[/B], [B]9997[/B] are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes)

**** None of 947, 977, 997, 9477, 9977 are primes.

* Case (9,B):

** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, [B]90B[/B], [B]9BB[/B] are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes)

*** Since 27, 77B, [B]929B[/B], [B]992B[/B], [B]997B[/B] are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes)

**** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes)

***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime)

***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime)

**** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes)

***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes)

****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime.

****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime)

****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime.

**** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B

***** None of 99B, 999B, 979B, 9799B, 9979B are primes.

* Case (A,1):

** Since A7, AB, 11, 31, 51, 61, 81, 91, [B]A41[/B] are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes)

*** Since 221, 2A1, [B]A0A1[/B], [B]A201[/B] are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes)

**** For the A{A}{0}1 family:

***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime.

***** The smallest prime of the form AA{0}1 is [B]AA000001[/B]

***** The smallest prime of the form AAA{0}1 is [B]AAA0001[/B]

***** The smallest prime of the form AAAA{0}1 is [B]AAAA1[/B]

****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc.

**** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime.

sweety439 2021-01-08 20:19

* Case (A,5):

** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes)

*** Since 565, 655, 665, [B]A605[/B], [B]A6A5[/B], [B]AA65[/B] are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes)

**** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime.

**** The smallest prime of the form A{0}65 is [B]A00065[/B]

* Case (A,7):

** [B]A7[/B] is prime, and thus the only minimal prime in this family.

* Case (A,B):

** [B]AB[/B] is prime, and thus the only minimal prime in this family.

* Case (B,1):

** Since B5, B7, 11, 31, 51, 61, 81, 91, [B]B21[/B] are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes)

*** Since 4B, AB, 401, A41, [B]B001[/B], [B]B0B1[/B], [B]BB01[/B], [B]BB41[/B] are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes)

**** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1

***** For the B0{4}{A}1 family, since [B]B04A1[/B] is prime, we only need to consider the families B0{4}1 and B0{A}1

****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime)

****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)

***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1

****** The smallest prime of the form B{A}01 is [B]BAA01[/B]

****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime)

**** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1

***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1

****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1

******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1

******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes.

****** For the B04{4,A}1 family, since [B]B04A1[/B] is prime, we only need to consider the family B04{4}1

******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime)

***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes)

****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime)

****** The smallest prime of the form B04{A}1 is [B]B04A1[/B]

****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime)

****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime)

**** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1

***** Since [B]BAAA1[/B] is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1

****** For the B{A,B}A1 family, since AB and [B]BAAA1[/B] are primes, we only need to consider the families B{B}A1 and B{B}AA1

******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime.

******* The smallest prime of the form B{B}AA1 is [B]BBBAA1[/B]

****** For the B{A,B}AA1 family, since [B]BAAA1[/B] is prime, we only need to consider the families B{B}AA1

******* The smallest prime of the form B{B}AA1 is [B]BBBAA1[/B]

**** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes)

***** The smallest prime of the form B{B}B1 is [B]BBBB1[/B]

***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime.

***** The smallest prime of the form B{B}BAA1 is [B]BBBAA1[/B]

* Case (B,5):

** [B]B5[/B] is prime, and thus the only minimal prime in this family.

* Case (B,7):

** [B]B7[/B] is prime, and thus the only minimal prime in this family.

* Case (B,B):

** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, [B]B2B[/B] are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes)

*** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B

**** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B

***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime.

***** For the B{0,B}9B family:

****** Since [B]B0B9B[/B] and [B]BB09B[/B] are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes)

******* The smallest prime of the form B{0}9B is [B]B0000000000000000000000000009B[/B]

******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1), thus cannot be prime.

***** For the B{0,B}99B family:

****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes)

******* The smallest prime of the form B{0}99B is [B]B00099B[/B]

******* The smallest prime of the form B{B}99B is [B]BBBBBB99B[/B]

***** For the B{0,B}999B family:

****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes)

******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B (not minimal prime, since B00099B and B0000000000000000000000000009B are primes)

******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B (not minimal prime, since BBBBBB99B is prime)

sweety439 2021-01-08 20:21

[QUOTE=sweety439;568654]base 13:

[CODE]
184: 2393
190: 917093711
196: 2549
202: 75001187
262: 3407
268: 45293
274: 46307
280: 615161
352: 10053473
358: 292031598119
382: 4967
388: 852437
406: 5279
412: 152972717
430: 5591
436: 5669
460: 5981
466: 13309427
[no known prime for S13 k=484, 2B3{0}1 is unsolved family in base 13]
490: 82811
538: 1181987
544: 1195169
574: 97007
580: 7541
616: 8009
622: 8087
652: 531856430093
658: 90710887636643
874: 264712843161629123
880: 148721
886: 11519
892: 11597
952: 12377
958: 2104727
964: 162917
970: 12611
1198: 34216079
1204: 2645189
1210: 15731
1216: 15809
1276: 80067107693
1282: 36615203
1288: 2829737
1294: 16823
1366: 652758153339229918146262170195611899412691250234661073608281535607441327245782154274226163627511856910242680997040239850166408242104102157590110059393064877741749052590786638302327896695517321882076385248095689607714279245947320040491619053213124101066946141643999231501774298529691499984819614064572813909251081765622950846989229615735750095118621279499641063473790506905622553548920945551313745769942678628213020977681299198153663054881959698284804196609230966508441216040759006656791607
1372: 17837
1396: 18149
1402: 12127883118972635782067
1420: 18461
1426: 18539
1444: 18773
1450: 41413451
1474: 19163
1480: 448255157756534441
1498: 34900531513476539
1504: 19553
1552: 20177
1558: 263303
1588: 2321529421116208423755077
1594: 269387
1630: 21191
1636: 21269
1666: 47582627
1672: 21737
1888: 821828298004735653739505427655167983139612951149032024600411489000613133621406169905393832026983244445467247164419724522787738176502646863025015739869100207704101192365412789485747731430993843624803368161051869616583490868130229284719241207489969244826144259775702398961583808046895149627729346180500126934284621986296138708001440861667299169
1894: 24623
1900: 1549888369901
1906: 4187483
1966: 4319303
1972: 333269
1978: 56493659
1984: 25793
[/CODE][/QUOTE]

484*13^15198+1 (2B30[SUB]15197[/SUB]1) is prime!!!

This prime is likely the third-largest "base 13 minimal prime (start with base+1)" (there is a larger probable prime 80[SUB]32017[/SUB]111, and there is an unsolved family in base 13: 9{5})

pinhodecarlos 2021-01-08 20:37

Have you thought about writing a book about all of this exciting stuff?

VBCurtis 2021-01-08 21:11

14 posts on a single topic in 36 hours. Are you trying to get banned? It's working.

sweety439 2021-01-09 10:34

1 Attachment(s)
[QUOTE=pinhodecarlos;568764]Have you thought about writing a book about all of this exciting stuff?[/QUOTE]

I have a pdf file for the proof (not complete, continue updating), I will complete the proof for bases 7, 9, 12

sweety439 2021-01-09 10:38

It is conjectured that for all simple families x{y}z cannot be proved as only contain composites (for numbers > base) in one of these four ways:

** Periodic sequence p of prime divisors with p(n) | (xyyy...yyyz with n y's)
** Algebraic factors (e.g. difference-of-squares factorization, difference-of-cubes factorization, sum-of-cubes factorization, difference-of-5th-powers factorization, sum-of-5th-powers factorization, Aurifeuillian factorization of x^4+4*y^4, etc.) of x{y}z
** The combine of the above two ways (like the case of {B}9B in base 12)
** Reduced to (b^(r*n+s)+1)/gcd(b+1,2), and r*n+s can never be power of 2 (like the case of 8{0}1 in base 128)

Then x{y}z contain primes (for numbers > base).

sweety439 2021-01-09 10:53

1 Attachment(s)
The simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b are of the form (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), this number has algebra factors if and only if:

either

* there is an integer r>1 such that both a*b^n and -c are perfect rth powers

or

* a*b^n*c is of the form 4*m^4 with integer m

If (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebra factors, then it must be composite, the only exception is when it is either GFN (generalized Fermat number) base b or GRU (generalized repunit number) base b, in these two cases this number may be prime, the only condition is the n is power of 2 if it is GFN, and the n is prime if it is GRU (since GFN and GRU are the only (a*b^n+c)/gcd(a+c,b-1) which is [URL="https://en.wikipedia.org/wiki/Divisibility_sequence"]divisibility sequence[/URL])

References for GRU (why n must be prime): [URL="https://www.mersenne.org/various/math.php"]https://www.mersenne.org/various/math.php[/URL] [URL="https://primes.utm.edu/mersenne/"]https://primes.utm.edu/mersenne/[/URL] [URL="https://en.wikipedia.org/wiki/Repunit"]https://en.wikipedia.org/wiki/Repunit[/URL] [URL="https://mathworld.wolfram.com/RepunitPrime.html"]https://mathworld.wolfram.com/RepunitPrime.html[/URL] [URL="https://oeis.org/A000043"]https://oeis.org/A000043[/URL] [URL="https://oeis.org/A004023"]https://oeis.org/A004023[/URL]

References for GFN (why n must be power of 2): [URL="http://www.worldofnumbers.com/deplat.htm"]http://www.worldofnumbers.com/deplat.htm[/URL] [URL="http://www.fermatsearch.org/math.html"]http://www.fermatsearch.org/math.html[/URL] [URL="https://en.wikipedia.org/wiki/Fermat_number"]https://en.wikipedia.org/wiki/Fermat_number[/URL] [URL="https://mathworld.wolfram.com/FermatNumber.html"]https://mathworld.wolfram.com/FermatNumber.html[/URL] [URL="https://oeis.org/A092506"]https://oeis.org/A092506[/URL]

Also see [URL="https://stdkmd.net/nrr/repunit/repunitnote.htm"]this page[/URL] and [URL="https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf"]this page[/URL]

When we sieve the sequence (a*b^n+c)/gcd(a+c,b-1) (where a>=1, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), we will remove the n such that either "(a*b^n+c)/gcd(a+c,b-1) has a prime factor < certain limit (e.g. 10^9 or 10^12)" or "(a*b^n+c)/gcd(a+c,b-1) has algebra factors", however, if (a*b^n+c)/gcd(a+c,b-1) is GFN, then we will remove all n-values except powers of 2, thus there are only few remain n-values < certain limit (e.g. 100K or 1M), thus it is no need to sieve, instead, we can use [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]trial division[/URL], e.g. [URL="http://www.prothsearch.com/fermat.html"]the trial division of Fermat numbers[/URL], and if (a*b^n+c)/gcd(a+c,b-1) is GRU, then we will remove all composite n (and leave all prime n) and use the sieve program (without removing the n with algebra factors, only remove the n with small prime factors) remove the n such that (a*b^n+c)/gcd(a+c,b-1) has a prime factor < certain limit (e.g. 10^9 or 10^12) instead of remove the n such that (a*b^n+c)/gcd(a+c,b-1) has algebra factors, since this will remove all n-values (see [URL="https://mersenneforum.org/showpost.php?p=452132&postcount=66"]this post[/URL] and [URL="https://www.mersenneforum.org/showthread.php?t=22740"]this thread[/URL]).

Form (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) is GFN or GRU iff c=+-1 and a is rational power of b

Form (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) is GFN iff:

* c=1, a = b^(r/s) (a is rational power of b), gcd(r,s) = 1, s is odd [case b is even is standard GFN, case b is odd is half GFN]

Form (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) is GRU iff:

* c=-1, a = b^(r/s) (a is rational power of b) [in this case the form is GRU to positive base]

or

* c=1, a = b^(r/s) (a is rational power of b), gcd(r,s) = 1, s is even [in this case the form is GRU to negative base]

In fact, this is equivalent to:

Form (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) is GFN iff:

* c=1, a is rational power of b, gcd(a+c,b-1) = 1 [in this case the form is standard GFN]

or

* c=1, a is rational power of b, gcd(a+c,b-1) = 2 [in this case the form is half GFN]

Form (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) is GRU iff:

* c=-1, a is rational power of b [in this case the form is GRU to positive base gcd(a+c,b-1)+1]

or

* c=1, a is rational power of b, gcd(a+c,b-1) > 2 [in this case the form is GRU to negative base -(gcd(a+c,b-1)-1)]

GFN and GRU are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b whose prime factors must be == 1 mod n for an integer n (by the theorem that all prime factors of cyclotomic number Phi(n,b) are either == 1 mod n or equal [URL="https://oeis.org/A006530"]lpf[/URL](n)) (e.g. [URL="https://www.mersenne.org/various/math.php"]all prime factors of Mersenne number ({1} in base 2) Mp = Phi(p,2) are == 1 mod p[/URL] (in fact, == 1 mod 2*p since 2 is [URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] mod all primes p == 7 mod 8) [URL="http://www.fermatsearch.org/algorithm.html"]all prime factors of Fermat number (1{0}1 in base 2) Fn = Phi(2^(n+1),2) are == 1 mod 2^(n+1)[/URL] (in fact, == 1 mod 2^(n+2) since 2 is [URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] mod all primes p == 1 mod 8)), thus when we sieve GFN and GRU, we only need to sieve the primes == 1 mod n (of course, our sieve list should only include power-of-2 n for GFN and only include prime n for GRU, since other n have algebra factors).

GFN and GRU are the only simple families x{y}z (where x and z are strings of base b digits, y is base b digit) in base b which are also cyclotomic numbers (i.e. numbers of the form Phi(n,b)/gcd(Phi(n,b),n), where Phi is [URL="https://en.wikipedia.org/wiki/Cyclotomic_polynomial"]cyclotomic polynomial[/URL]) or Zsigmondy numbers Zs(n,b,1) (see [URL="https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem"]Zsigmondy's theorem[/URL])

By the definition of [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL], GFN and GRU have difficulty zero, and they are the only simple families x{y}z with no covering set (including: full numerical covering set, full algebraic covering set, partial numerical/partial algebraic covering set) but have difficulty zero, thus, they are also the only simple families x{y}z which cannot be proven to contain no primes > base, but have difficulty zero.

(references of examples of difficulty calculating: [URL="https://stdkmd.net/nrr/1/11113.htm#prime_period"]https://stdkmd.net/nrr/1/11113.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/13333.htm#prime_period"]https://stdkmd.net/nrr/1/13333.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/10003.htm#prime_period"]https://stdkmd.net/nrr/1/10003.htm#prime_period[/URL])

All GFN base b and all GRU base b are strong-probable-primes (primes and [URL="https://en.wikipedia.org/wiki/Strong_pseudoprime"]strong pseudoprimes[/URL]) to base b, since they are over-probable-primes (primes and overpseudoprimes) to base b (references: [URL="https://oeis.org/A141232"]https://oeis.org/A141232[/URL] [URL="http://arxiv.org/abs/0806.3412"]http://arxiv.org/abs/0806.3412[/URL] [URL="http://arxiv.org/abs/0807.2332"]http://arxiv.org/abs/0807.2332[/URL] [URL="http://arxiv.org/abs/1412.5226"]http://arxiv.org/abs/1412.5226[/URL] [URL="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.pdf[/URL]), and all overpseudoprimes are [URL="https://en.wikipedia.org/wiki/Strong_pseudoprime"]strong pseudoprimes[/URL] to the same base b, all strong pseudoprimes are [URL="https://en.wikipedia.org/wiki/Euler%E2%80%93Jacobi_pseudoprime"]Euler–Jacobi pseudoprimes[/URL] to the same base b, all Euler–Jacobi pseudoprimes are [URL="https://en.wikipedia.org/wiki/Euler_pseudoprime"]Euler pseudoprimes[/URL] to the same base b, all Euler pseudoprimes are [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprimes[/URL] to the same base b, so don't test with this base (see [URL="https://mersenneforum.org/showthread.php?t=10476&page=2"]https://mersenneforum.org/showthread.php?t=10476&page=2[/URL], [URL="https://mersenneforum.org/showpost.php?p=483302&postcount=85"]https://mersenneforum.org/showpost.php?p=483302&postcount=85[/URL], [URL="https://mersenneforum.org/showpost.php?p=611607&postcount=10"]https://mersenneforum.org/showpost.php?p=611607&postcount=10[/URL], [URL="https://oeis.org/A171381"]https://oeis.org/A171381[/URL], [URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL], also see [URL="https://oeis.org/A210454"]https://oeis.org/A210454[/URL], [URL="https://oeis.org/A210461"]https://oeis.org/A210461[/URL], [URL="https://oeis.org/A216170"]https://oeis.org/A216170[/URL], [URL="https://oeis.org/A217841"]https://oeis.org/A217841[/URL], [URL="https://oeis.org/A243292"]https://oeis.org/A243292[/URL], [URL="https://oeis.org/A217853"]https://oeis.org/A217853[/URL], [URL="https://oeis.org/A293626"]https://oeis.org/A293626[/URL], [URL="https://oeis.org/A210454/a210454.pdf"]https://oeis.org/A210454/a210454.pdf[/URL], [URL="https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf[/URL], all generalized repunits in base b^2 with length p (where p is prime not dividing b*(b^2-1)) are Fermat pseudoprimes to base b, thus there are infinitely many pseudoprimes to every base b), in fact, all [URL="https://oeis.org/A323748"]Zsigmondy numbers[/URL] Zs(n,b,1) ant all their factors are strong-probable-primes (primes and strong pseudoprimes) to base b, since they are over-probable primes (primes and overpseudoprimes) to base b, so don't test with this base. ([URL="http://ntheory.org/pseudoprimes.html"]reference of list of pseudoprimes[/URL])

GFNs and GRUs are [URL="https://en.wikipedia.org/wiki/Cunningham_number"]Cunningham numbers[/URL] ([URL="https://mathworld.wolfram.com/CunninghamNumber.html"]Mathworld[/URL] [URL="https://oeis.org/A080262"]OEIS sequence[/URL]), i.e. of the form b^n+-1 with b>=2, n>=2, if C+(b,n) = b^n+1 is prime, then b is even number and n is power of 2, and if C-(b,n) = b^n-1 is prime, then b = 2 and n is prime, if we take out the trivial factor (like we [URL="https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub"]take out the trivial factor gcd(k+1,b-1) for k*b^n+1 in generalized Sierpinski problem base b[/URL] and [URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]take out the trivial factor gcd(k-1,b-1) for k*b^n-1 in generalized Riesel problem base b[/URL]), we take out trivial factor 2 from C+(b,n) = b^n+1 for odd b and take out trivial factor b-1 from C-(b,n) = b^n-1 for b>2, thus, C+(b,n) = b^n+1 become (b^n+1)/gcd(b-1,2) (exactly the GFN formula) and C-(b,n) = b^n-1 become (b^n-1)/(b-1) (exactly the GRU formula), they have the same properties as original b^n+-1, i.e. if (b^n+1)/gcd(b-1,2) is prime, then n is power of 2, and if (b^n-1)/(b-1) is prime, then n is prime.

GFNs and GRUs in bases 2<=b<=64:

[CODE]
base GFN family GRU family
2 1{0}1 {1}
3 {1}2 {1}
4 1{0}1 1{3}, {2}3
5 {2}3 {1}
6 1{0}1 {1}
7 {3}4 {1}
8 2{0}1, 4{0}1 1{7}, 3{7}
9 {4}5 1{4}, {6}7
10 1{0}1 {1}
11 {5}6 {1}
12 1{0}1 {1}
13 {6}7 {1}
14 1{0}1 {1}
15 {7}8 {1}
16 1{0}1 1{F}, 7{F}, {A}B, 2{A}B
17 {8}9 {1}
18 1{0}1 {1}
19 {9}A {1}
20 1{0}1 {1}
21 {A}B {1}
22 1{0}1 {1}
23 {B}C {1}
24 1{0}1 {1}
25 {C}D 1{6}, {K}L
26 1{0}1 {1}
27 1{D}E, 4{D}E 1{D}, 4{D}
28 1{0}1 {1}
29 {E}F {1}
30 1{0}1 {1}
31 {F}G {1}
32 2{0}1, 4{0}1, 8{0}1, G{0}1 1{V}, 3{V}, 7{V}, F{V}
33 {G}H {1}
34 1{0}1 {1}
35 {H}I {1}
36 1{0}1 1{7}, {U}V
37 {I}J {1}
38 1{0}1 {1}
39 {J}K {1}
40 1{0}1 {1}
41 {K}L {1}
42 1{0}1 {1}
43 {L}M {1}
44 1{0}1 {1}
45 {M}N {1}
46 1{0}1 {1}
47 {N}O {1}
48 1{0}1 {1}
49 {O}P 1{8}, {g}h
50 1{0}1 {1}
51 {P}Q {1}
52 1{0}1 {1}
53 {Q}R {1}
54 1{0}1 {1}
55 {R}S {1}
56 1{0}1 {1}
57 {S}T {1}
58 1{0}1 {1}
59 {T}U {1}
60 1{0}1 {1}
61 {U}V {1}
62 1{0}1 {1}
63 {V}W {1}
64 4{0}1, G{0}1 1{$}, V{$}, {g}h, A{g}h
[/CODE]

The smallest prime (single-digit primes are not counted) in this families are in the text file.

Note: we do not include the case where the "ground base" of the GFNs is perfect odd power and the case where the "ground base" of the GRUs is either perfect power or of the form -4*m^4 with integer m, since such numbers have algebra factors and are composite for all n or are prime only for very small n, such families only exist in perfect odd power bases for the GFNs and perfect power bases for the GRUs (case -4*m^4 only exists in perfect 4th power bases), such families for bases 2<=b<=64 are:

[CODE]
base GFN family GRU family
4 {1}
8 1{0}1 {1}
9 {1}
16 {1}, 1{5}, {C}D
25 {1}
27 {D}E {1}
32 1{0}1 {1}
36 {1}
49 {1}
64 1{0}1 {1}, 1{L}, 5{L}, 1{9}, {u}v
[/CODE]

Such small primes are: 11 in base 4, 111 in base 8, 11 in base 16, 111 in base 27, 11 in base 36, 19 in base 64

Note: the "ground base" of the GFNs or GRUs need not to be b (when b is perfect power), it may be root of b, it may also be negative integer which is root of b

sweety439 2021-01-09 21:18

These bases 2<=b<=1024 have unsolved families which are GFNs:

{31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024}

Such families are:

* 4:{0}:1, 16:{0}:1 for b = 32
* 12:{62}:63 for b = 125
* 16:{0}:1 for b = 128
* 36:{0}:1 for b = 216
* 24:{171}:172 for b = 343
* 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512
* 10:{0}:1, 100:{0}:1 for b = 1000
* 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024
* 1:{0}:1 for other even bases b
* {((b-1)/2)}:((b+1)/2) for other odd bases b

These bases 2<=b<=1024 have unsolved families which are GRUs:

{185, 243, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015}

Such families are:

* 40:{121} for b = 243
* {1} for other bases b

carpetpool 2021-01-09 21:44

Here's a suggestion:

Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space!

The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes.

Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?

sweety439 2021-01-10 12:09

[QUOTE=carpetpool;568862]Here's a suggestion:

Instead of constantly posting search limits and reservations that can be done in minutes, use a doc or pdf instead. Attach or provide the link in a single post. If you need to edit the link or update the attachment, edit the post instead of creating a new post. It saves time and space!

The updates you see on CRUS or other prime search projects take at least several months, if not years to complete. Now your searches and updates on the other hand, can be done in minutes.

Why post something that is so trivial that anyone who wants to do it can do it in such a short amount of time?[/QUOTE]

I have made a pdf file about this project, see the attached file in post [URL="https://mersenneforum.org/showpost.php?p=568813&postcount=114"]#114[/URL]

Well, the proofs for base 2, 3, 4 are really trivial, but they are part of the project, I want to store these proofs, and the pdf file was made recently

sweety439 2021-01-10 19:27

Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=12:

[CODE]
11
15
17
1B
25
27
31
35
37
3B
45
4B
51
57
5B
61
67
6B
75
81
85
87
8B
91
95
A7
AB
B5
B7
221
241
2A1
2B1
2BB
401
421
447
471
497
565
655
665
701
70B
721
747
771
77B
797
7A1
7BB
907
90B
9BB
A41
B21
B2B
2001
200B
202B
222B
229B
292B
299B
4441
4707
4777
6A05
6AA5
729B
7441
7B41
929B
9777
992B
9947
997B
9997
A0A1
A201
A605
A6A5
AA65
B001
B0B1
BB01
BB41
600A5
7999B
9999B
AAAA1
B04A1
B0B9B
BAA01
BAAA1
BB09B
BBBB1
44AAA1
A00065
BBBAA1
AAA0001
B00099B
AA000001
BBBBBB99B
B0000000000000000000000000009B
400000000000000000000000000000000000000077
[/CODE]

sweety439 2021-01-10 19:28

There are totally 106 minimal primes (start with 2 digits) in base 12, there are 77 such primes in base 10

sweety439 2021-01-10 20:09

1 Attachment(s)
All known minimal primes (start with b+1) in bases 2<=b<=16: (data for bases 2, 3, 4, 5, 6, 8, 10, 12 are known to be complete)

Also three unsolved families are known:

Base 11: 5777...777
Base 13: 9555...555
Base 16: DBBB...BBB

Uncwilly 2021-01-10 20:47

Rather than spilling out each new thought that comes to your head when it does. Try writing them down off line and posting only 1 well formatted post every couple of days. Again think about formatting it nicely in your word processor and then making a PDF that you can later update.

I am tired of seeing you posting drival serval times a day.

sweety439 2021-01-10 22:21

1 Attachment(s)
fixed typo in the file: 3*16^n+1 should be 3*4^n+1, 12^((n+1)/2) +/- 5 should be 12^((n+2)/2) +/- 5

VBCurtis 2021-01-10 22:45

You have an "edit" button. Use it. ONE pdf for this crap, not a new post with a new PDF every time you change a sentence. Edit the post, edit the PDF

If you were limited to one post per day on the forum, would this be it?

I'm deleting your first PDF and its post.

sweety439 2021-01-12 18:50

[QUOTE=sweety439;567702]* The smallest Williams prime with 4th kind base b (for b != 1 mod 3): [not minimal prime (start with 2 digits) if either b is prime or base b has smaller generalized Fermat prime, but for the case that b is prime, it is still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

7, 13, 31, 43, 73, 811, 1453, 157, 211, 241, 307, 3768826516993, 421, 463, 12697, 601, 18253, 757, 615334471, 27901, 1107296257, 1123, 44101, 1726273, 1483, 2372761, 1723, 75853, 87121, 93151, 106033, 599298932737, 2551, 158981126352779044590102826209115342318059775372698133871491241388097301966680877821738760704616125782843491355455960710073030287313404870590681666644752545879191893959727029866211537628677981607279205572507381073830401006677162824033234341436459420880686565908174585159142942438136179315586329074318947952541865853, 151687, 2971, 178753, 3307, 3541, 1338153989063049216000000000000000000001, 3907, 48326086052867645032352571108528903615254734667108057821332757600957454538355546211631290156513879123036351230974951391062798157776810891656336682957284917485088940693788242992185798654992956966627018064055387274320725152943868432582696386314597516885379356294528772183874293272350708412107233383892387582454781698467578958840732553153[/QUOTE]

* The smallest prime of the form 2*b^n+1 (for b != 1 mod 3): [not minimal prime (start with 2 digits) if b<=2]

5, 7, 11, 13, 17, 19, 23, 3457, 29, 31, 13555929465559461990942712143872578804076607708197374744547, 37, 41, 43, 47, 1153, 53, 1459, 59, 61, 65537, 67, 71, 73, 2*38^2729+1, 79, 83, 3529, 89, 4051, 82823796591884729837907950243851987042491027688029791782033968173988787397927431168748344242980462637086843228831225333542602440512725127029105275975234384910715377295392116427292929375082823988662090607733781357479215392846048752706418227733688234263166843856633793191822664770551012658601887, 97, 101, 103, 107, 109, 113, 370387, 410759, 432001, 236522599840432068647134316649762315445236710001482847056204302486382634336257, 127

* The smallest prime of the form 2*b^n-1: [not minimal prime (start with 2 digits) if b<=2]

3, 5, 7, 1249, 11, 13, 127, 17, 19, 241, 23, 337, 76831, 29, 31, 577, 647, 37, 20479999999999, 41, 43, 296071777, 47, 1249, 617831551, 53, 1567, 15387133080032326246081223292828787411221911122916017220126284227825703776392672467768318856009763825207593900596158761682711294895921233392537083406917227083982402321012446032594528728383203531755841, 59, 61, 2147483647, 2582935937, 67, 3676531249, 71, 73, 2887, 3041, 79, 3361, 83, 3697, 7496191, 89, 4231, 9759361, 10616831, 97, 4999, 101, 103, 14762783749438524018088313240622157671545425891033638774020213131211643094561, 107, 109, 6271, 113, 390223, 6961, 1555199999, 453961, 7687, 7937, 127

* The smallest prime of the form b^n+2 (for b == 3, 5 mod 6): [not minimal prime (start with 2 digits) if either b<=2 or b is prime, but still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

5, 7, 11, 13, 17, 19, 23, 952809757913929, 29, 31, 1091, 37, 41, 43, 47, 885233716287722386108568808645559198522547790058305212262181780420828956357982973084581935827930464156048602918053397761948271781610736426217362565287242033121579185919812362859356307201329, 53, 1174711139839, 59, 61, 250049

* The smallest prime of the form b^n-2 (for odd b): [not minimal prime (start with 2 digits) if b<=2]

7, 3, 5, 7, 14639, 11, 13, 24137567, 17, 19, 480250763996501976790165756943039, 23, 727, 839, 29, 31, 1223, 1367, 37, 2825759, 41, 43, 2207, 47, 45767944570399, 7890479, 53, 1176246293903439667999, 12117359, 59, 61

* The smallest prime of the form 3*b^n+1 (for even b): [not minimal prime (start with 2 digits) if b<=3]

7, 13, 19, 193, 31, 37, 43, 769, 17497, 61, 67, 73, 79, 40478785537, 81001, 97, 103, 109, 164617, 4801, 127, 1854365518528513, 139, 331777, 151, 157, 163, 1652195329, 10093, 181, 9678800287193699463169, 193

* The smallest prime of the form 3*b^n-1 (for even b): [not minimal prime (start with 2 digits) if b<=3]

5, 11, 17, 23, 29, 431, 41, 47, 53, 59, 1451, 71, 2027, 83, 89, 108086391056891903, 101, 107, 113, 4799, 3*42^2523-1, 131, 137, 6911, 149, 8111, 8747, 167, 173, 179, 16913400588503030024793898903900960521239102670648159766677517992069347477035908686646316997626793866297343, 191

* The smallest prime of the form b^n+3 (for b == 2, 4 mod 6): [not minimal prime (start with 2 digits) if either b<=3 or b is prime, but still minimal prime (start with 2 digits) if we use LaurV's suggestion, i.e. start with b+1 instead of b]

5, 7, 11, 13, 17, 19, 23, 487, 29, 31, 32771, 37, 41, 43, 47, 1799519816997495209117766334283779, 53, 2707, 59, 61, 3847, 67

* The smallest prime of the form b^n-3 (for b == 2, 4 mod 6): [not minimal prime (start with 2 digits) if b<=3]

5, 13, 5, 7, 11, 13, 17, 19, 23, 296196766695421, 29, 31, 54869, 37, 41, 43, 47, 1514785299052682515540398802570879414320893571359760514960122067313271212237031712057484726921232170496646835505906834446399053647478565523037279529736578428914328808517619293356029, 53, 3361, 59, 61

sweety439 2021-01-12 18:59

[QUOTE=sweety439;567582]Base b minimal primes (start with 2 digits) includes:

* The smallest repunit prime base b if exists
* The smallest generalized Fermat prime base b for even b if exists
* The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists
* The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]Williams prime with 1st kind[/URL] base b if exists
* The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]Williams prime with 2nd kind[/URL] base b if exists
* The smallest [URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]Williams prime with 4th kind[/URL] base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest dual Williams prime with 1st kind base b if exists
* The smallest dual Williams prime with 2nd kind base b [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest dual Williams prime with 4th kind base b [B]for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest prime of the form 2*b^n+1 for bases b>2 if exists
* The smallest prime of the form 2*b^n-1 for bases b>2 if exists
* The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists
* The smallest prime of the form 3*b^n+1 for bases b>3 if exists
* The smallest prime of the form 3*b^n-1 for bases b>3 if exists
* The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists
* The smallest prime of the form 4*b^n+1 for bases b>4 if exists
* The smallest prime of the form 4*b^n-1 for bases b>4 if exists
* The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 [B]for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded)[/B] if exists
* The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists
...
* The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]CRUS Sierpinski conjecture[/URL] for fixed 1<=k<=b-1) if exists
* The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]CRUS Riesel conjecture[/URL] for fixed 1<=k<=b-1) if exists
* The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and [B]gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...)[/B] (i.e. the prime for the [URL="https://mersenneforum.org/attachment.php?attachmentid=24053&d=1609098432"]extended Riesel conjecture[/URL] for fixed k satisfying these two conditions) if exists
* The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1[/QUOTE]

The corresponding families:

* repunit prime base b: {1}
* generalized Fermat prime base b for even b: 1{0}1
* generalized half Fermat prime (> (b+1)/2) base b for odd b: {x}y, x = (b-1)/2, y = (b+1)/2
* Williams prime with 1st kind base b: x{y}, x = b-2, y = b-1
* Williams prime with 2nd kind base b: x{0}1, x = b-1
* Williams prime with 4th kind base b: 11{0}1 [B](not minimal prime if there is smaller prime of the form 1{0}1[/B]
* dual Williams prime with 1st kind base b: {x}1, x = b-1
* dual Williams prime with 2nd kind base b: 1{0}x, x = b-1
* dual Williams prime with 4th kind base b: 1{0}11 [B](not minimal prime if there is smaller prime of the form 1{0}1[/B]
* prime of the form 2*b^n+1 for bases b>2: 2{0}1
* prime of the form 2*b^n-1 for bases b>2: 1{x}, x = b-1
* prime of the form b^n+2 for bases b>2 with gcd(b,2)=1: 1{0}2
* prime of the form b^n-2 for bases b>2 with gcd(b,2)=1: {x}y, x = b-1, y = b-2
* prime of the form 3*b^n+1 for bases b>3: 3{0}1
* prime of the form 3*b^n-1 for bases b>3: 2{x}, x = b-1
* prime of the form b^n+3 for bases b>3 with gcd(b,3)=1: 1{0}3
* prime of the form b^n-3 for bases b>3 with gcd(b,3)=1: {x}y, x = b-1, y = b-3
* prime of the form 4*b^n+1 for bases b>4: 4{0}1
* prime of the form 4*b^n-1 for bases b>4: 3{x}, x = b-1
* prime of the form b^n+4 for bases b>4 with gcd(b,4)=1: 1{0}4
* prime of the form b^n-4 for bases b>4 with gcd(b,4)=1: {x}y, x = b-1, y = b-4
* prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Sierpinski conjecture for fixed 1<=k<=b-1): k{0}1
* prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Riesel conjecture for fixed 1<=k<=b-1): x{y}, x = k-1, y = b-1
* prime of the form b^n+k for fixed 1<=k<=b-1: 1{0}k
* prime of the form b^n-k for fixed 1<=k<=b-1: {x}y, x = b-1, y = b-k
* prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1: x{y}, x = (k-1)/gcd(k-1,b-1), y = (b-1)/gcd(k-1,b-1)
* prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1: x = (b-1)/gcd(k-1,b-1), y = (b-k)/gcd(k-1,b-1)

sweety439 2021-01-12 19:03

This puzzle is an extension of the original [URL="https://www.primepuzzles.net/puzzles/puzz_178.htm"]minimal prime base b puzzle[/URL], to include CRUS [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski[/URL]/[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel[/URL] conjectures base b with k-values < b, i.e. the smallest prime of the form k*b^n+1 and k*b^n-1 for all k < b

Also include the dual Sierpinski/Riesel conjectures (of course in the dual case, gcd(k,b) = 1 is needed) base b with k-values < b, i.e. the smallest prime of the form b^n+k and b^n-k for all k < b

This problem is finding the minimal set of the set of [B]primes > b[/B] in base b, for bases 2<=b<=36, [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]the original minimal prime problem[/URL] is finding the minimal set of the set of [B]primes[/B] in base b, for bases 2<=b<=30.

This problem is better than [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]the original minimal prime problem[/URL] since this problem does not regard whether 1 is considered as prime or not. (in fact, if 1 is considered as prime, then the original minimal prime problem is solved in all bases 2<=b<=24 except b=21, if [URL="https://primes.utm.edu/glossary/xpage/PRP.html"]probable primes[/URL] are allowed)

Also, this problem includes finding the smallest generalized [URL="https://primes.utm.edu/top20/page.php?id=15"]near-repdigit prime[/URL] of given form (xyyy...yyy or xxx...xxxy, where x and y are base-b digits) in base b (or proving that such prime does not exist), if the repeating digit (i.e. y for xyyy...yyy, or x for xxx...xxxy) is not 1 (while the original minimal prime problem does not include this, if x or y (or both) is prime), for the smallest generalized near-repdigit prime of given form (xyyy...yyy or xxx...xxxy, where x and y are base-b digits) in bases 2<=b<=36 (including the case where the repeating digit is 1, which do not give minimal prime (start with b+1) in base b unless base b has no generalized [URL="https://primes.utm.edu/glossary/xpage/Repunit.html"]repunit[/URL] primes, such bases 2<=b<=36 are 9, 25, 32), see [URL="https://raw.githubusercontent.com/xayahrainie4793/non-single-digit-primes/main/smallest%20generalized%20near-repdigit%20prime.txt"]https://raw.githubusercontent.com/xayahrainie4793/non-single-digit-primes/main/smallest%20generalized%20near-repdigit%20prime.txt[/URL] (this list skips the case where xyyy...yyy or xxx...xxxy has NUMERICAL covering set, 0 if either xyyy...yyy or xxx...xxxy has full covering set with all or partial ALGEBRAIC factors or no (probable) primes of the form xyyy...yyy or xxx...xxxy are known)

(in families below, we assume the repeating digit (i.e. y for x{y}, or x for {x}y, etc.) is not 1, unless the base has no repunit primes (such bases are 9, 25, 32, 49, 64, 81, 121, 125, 144, ... ([URL="https://oeis.org/A096059"]https://oeis.org/A096059[/URL])))

The smallest prime in these types of simple families (if exists) are [I]always[/I] minimal primes (start with b+1):

* x{y}
* {x}y
* x{0}y

If family x{y} can be ruled out as only contain composites, then the smallest prime in these simple families (if exists) are [I]always[/I] minimal primes:

* xx{y}
* x0{y}
* x{y}0y

If family {x}y can be ruled out as only contain composites, then the smallest prime in these simple families (if exists) are [I]always[/I] minimal primes:

* {x}yy
* {x}0y
* x0{x}y

If family x{0}y can be ruled out as only contain composites, then the smallest prime in these simple families (if exists) are [I]always[/I] minimal primes:

* xx{0}y
* xy{0}y
* x{0}yy
* x{0}xy

sweety439 2021-01-12 19:09

[QUOTE=sweety439;569091]* The smallest prime of the form b^n-2 (for odd b):

7, 3, 5, 7, 14639, 11, 13, 24137567, 17, 19, 480250763996501976790165756943039, 23, 727, 839, 29, 31, 1223, 1367, 37, 2825759, 41, 43, 2207, 47, 45767944570399, 7890479, 53, 1176246293903439667999, 12117359, 59, 61[/QUOTE]

The b^n-2 case should require n>1, since single-digit primes are not acceptable in this puzzle, thus the smallest primes should be:

7, 23, 47, 79, 14639, 167, 223, 24137567, 359, 439, 480250763996501976790165756943039, 6103515623, 727, 839, 29789, 1087, 1223, 1367, 2313439, 2825759, 1847, 1532278301220703123, 2207, 2399, 45767944570399, 7890479, 3023, 1176246293903439667999, 12117359, 3719, 3967

and the OEIS sequence for the exponent (n) should be [URL="https://oeis.org/A250200"]A250200[/URL], not [URL="https://oeis.org/A255707"]A255707[/URL]

Also the b^2-3 case (also should require n>1):

5, 13, 61, 97, 193, 4093, 397, 113379901, 673, 296196766695421, 1021, 1153, 54869, 1597, 1933, 2113, 476837158203124999999999999999999997, 1514785299052682515540398802570879414320893571359760514960122067313271212237031712057484726921232170496646835505906834446399053647478565523037279529736578428914328808517619293356029, 29334891491018187280695810850813, 3361, 916132829, 4093

sweety439 2021-01-13 15:57

[QUOTE=sweety439;568930]All known minimal primes (start with b+1) in bases 2<=b<=16: (data for bases 2, 3, 4, 5, 6, 8, 10, 12 are known to be complete)

Also three unsolved families are known:

Base 11: 5777...777
Base 13: 9555...555
Base 16: DBBB...BBB[/QUOTE]

DB[SUB]32234[/SUB] (base 16) is probable prime!!!

Its formula is (206*16^32234-11)/15

This number is like the largest minimal prime (start with 2 digits) in base 16

The families 5{7} (base 11) and 9{5} (base 13) still no (probable) prime found.

The formulas of these two families are (57*11^n-7)/10 and (113*13^n-5)/12, respectively.

sweety439 2021-01-13 16:48

1 Attachment(s)
Update newest pdf file.

Uncwilly 2021-01-13 16:58

[QUOTE=sweety439;569180]Update newest pdf file.[/QUOTE]
Since you have mod rights in this area, you can delete the previous PDF and replace it with the current. Keeping an up to date file and the first post of the thread is a common way of handling things like this. There is no need to post about a new item that will then be added to the pdf. Just update the file. Then maybe 1 time a week give a one line summary for each type of item updated. Less stuff to search through and better organization might make this more useful.

sweety439 2021-01-14 01:21

1 Attachment(s)
Update newest data file.

See [URL="https://github.com/xayahrainie4793/non-single-digit-primes"]https://github.com/xayahrainie4793/non-single-digit-primes[/URL] for more data.

sweety439 2021-01-14 03:53

These are families I am interested: (of the form (a*b^n+c)/gcd(a+c,b-1) for fixed a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 and variable n) (although some of these families do not always product minimal primes (start with b+1))

(for bases 2<=b<=1024)

* (b^n-1)/(b-1)
* b^n+1 for b == 0 mod 2
* (b^n+1)/2 for b == 1 mod 2
* b^n+2 for b == 3, 5 mod 6
* (b^n+2)/3 for b == 1 mod 6
* b^n+3 for b == 2, 4 mod 6
* (b^n+3)/2 for b == 1, 5 mod 6
* b^n+4 for b == 3, 5, 7, 9 mod 10 (b not == 14 mod 15, b not perfect 4th power)
* (b^n+4)/5 for b == 1 mod 10 (b not perfect 4th power)
* b^n-2 for b == 1 mod 2
* b^n-3 for b == 2, 4 mod 6
* (b^n-3)/2 for b == 1, 5 mod 6
* b^n-4 for b == 3, 5 mod 6 (b not == 4 mod 5, b not perfect square)
* (b^n-4)/3 for b == 1 mod 6 (b not perfect square)
* 2*b^n+1 for b == 0, 2 mod 3
* (2*b^n+1)/3 for b == 1 mod 3
* 3*b^n+1 for b == 0 mod 2
* (3*b^n+1)/2 for b == 1 mod 2
* 4*b^n+1 for b == 0, 2, 3, 4 mod 5 (b not == 14 mod 15, b not perfect 4th power)
* (4*b^n+1)/5 for b == 1 mod 5 (b not perfect 4th power)
* 2*b^n-1
* 3*b^n-1 for b == 0 mod 2
* (3*b^n-1)/2 for b == 1 mod 2
* 4*b^n-1 for b == 0, 2 mod 3 (b not == 4 mod 5, b not perfect square)
* (4*b^n-1)/3 for b == 1 mod 3 (b not == 4 mod 5, b not perfect square)

* b^n+5
* b^n+6
* b^n+7
* b^n+8
* b^n+9
* b^n+10
* b^n+11
* b^n+12
* b^n+13
* b^n+14
* b^n+15
* b^n+16
* b^n-5
* b^n-6
* b^n-7
* b^n-8
* b^n-9
* b^n-10
* b^n-11
* b^n-12
* b^n-13
* b^n-14
* b^n-15
* b^n-16

* 5*b^n+1
* 6*b^n+1
* 7*b^n+1
* 8*b^n+1
* 9*b^n+1
* 10*b^n+1
* 11*b^n+1
* 12*b^n+1
* 13*b^n+1
* 14*b^n+1
* 15*b^n+1
* 16*b^n+1
* 5*b^n-1
* 6*b^n-1
* 7*b^n-1
* 8*b^n-1
* 9*b^n-1
* 10*b^n-1
* 11*b^n-1
* 12*b^n-1
* 13*b^n-1
* 14*b^n-1
* 15*b^n-1
* 16*b^n-1

* 2*b^n+3
* 2*b^n-3
* 3*b^n+2
* 3*b^n-2
* 3*b^n+4
* 3*b^n-4
* 4*b^n+3
* 4*b^n-3

* {1}2 in base b
* {1}3 in base b
* {1}4 in base b
* {2}1 in base b
* {2}3 in base b
* {3}1 in base b
* {3}2 in base b
* {3}4 in base b
* {4}1 in base b
* {4}3 in base b
* 1{2} in base b
* 1{3} in base b
* 1{4} in base b
* 2{1} in base b
* 2{3} in base b
* 3{1} in base b
* 3{2} in base b
* 3{4} in base b
* 4{1} in base b
* 4{3} in base b

* (b/2)*b^n+1 for b == 0, 2 mod 6
* (b/2)*b^n-1 for b == 0 mod 2
* (3*b/2)*b^n+1 for b == 0, 2, 4, 8 mod 10
* (3*b/2)*b^n-1 for b == 0 mod 2
* (b/3)*b^n+1 for b == 0 mod 6
* (b/3)*b^n-1 for b == 0 mod 6
* (2*b/3)*b^n+1 for b == 0, 3, 9, 12 mod 15
* (2*b/3)*b^n-1 for b == 0 mod 3
* (4*b/3)*b^n+1 for b == 0, 3, 6, 9, 12, 18 mod 21
* (4*b/3)*b^n-1 for b == 0 mod 3
* (b/4)*b^n+1 for b == 0, 4, 8, 12 mod 20 (b not == 14 mod 15, b not perfect 4th power)
* (b/4)*b^n-1 for b == 0, 8 mod 12 (b not == 4 mod 5, b not perfect square)
* (3*b/4)*b^n+1 for b == 0, 4, 12, 16, 20, 24 mod 28
* (3*b/4)*b^n-1 for b == 0 mod 4

* b^n+(b-1)
* b^n-(b-1)
* b^n+(b+1) for b == 0, 2 mod 3
* (b^n+(b+1))/3 for b == 1 mod 3
* b^n-(b+1)
* (b-1)*b^n+1
* (b-1)*b^n-1
* (b+1)*b^n+1 for b == 0, 2 mod 3
* ((b+1)*b^n+1)/3 for b == 1 mod 3
* (b+1)*b^n-1
* (b^n+(b-2))/(b-1)
* ((b-2)*b^n+1)/(b-1)
* (b^n-(2*b-1))/(b-1)
* ((2*b-1)*b^n-1)/(b-1)
* (b-2)*b^n-1 for b == 0 mod 2
* (b+2)*b^n+1 for b == 0 mod 2
* (b+2)*b^n-1 for b == 0 mod 2
* (b*(b^2)^n+1)/(b+1) [this is the special case, original form is (b^n+1)/(b+1), but we should write the family as standard form ((a*b^n+c)/gcd(a+c,b-1) for fixed a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 and variable n)]

sweety439 2021-01-14 04:50

[CODE]
base (b) largest known minimal prime (start with b+1) in base b form largest known minimal prime (start with b+1) in algebraic ((a*b^n+c)/d) form length of largest known minimal prime (start with b+1) in base b form
2 11 3 [URL="http://factordb.com/index.php?showid=3&base=2"]2[/URL]
3 111 13 [URL="http://factordb.com/index.php?showid=13&base=3"]3[/URL]
4 221 41 [URL="http://factordb.com/index.php?showid=41&base=4"]3[/URL]
5 1(0^93)13 5^95+8 [URL="http://factordb.com/index.php?showid=1100000000034686071&base=5"]96[/URL]
6 40041 5209 [URL="http://factordb.com/index.php?showid=5209&base=6"]5[/URL]
7 (3^16)1 (7^17-5)/2 [URL="http://factordb.com/index.php?showid=116315256993601&base=7"]17[/URL]
8 (4^220)7 (4*8^221+17)/7 [URL="http://factordb.com/index.php?showid=1100000000416605822&base=8"]221[/URL]
9 3(0^1158)11 3*9^1160+10 [URL="http://factordb.com/index.php?showid=1100000002376318423&base=9"]1161[/URL]
10 5(0^28)27 5*10^30+27 [URL="http://factordb.com/index.php?showid=1100000000204142046&base=10"]31[/URL]
11 55(7^1011) (607*11^1011-7)/10 [URL="http://factordb.com/index.php?showid=1100000002361376522&base=11"]1013[/URL]
12 4(0^39)77 4*12^41+91 [URL="http://factordb.com/index.php?showid=1100000002375054575&base=12"]42[/URL]
13 8(0^32017)111 8*13^32020+183 [URL="http://factordb.com/index.php?showid=1100000000490878060&base=13"]32021[/URL]
14 4(D^19698) 5*14^19698-1 [URL="http://factordb.com/index.php?showid=1100000000884560233&base=14"]19699[/URL]
15 (7^155)97 (15^157+59)/2 [URL="http://factordb.com/index.php?showid=1100000002454891840&base=15"]157[/URL]
16 D(B^32234) (206*16^32234-11)/15 [URL="http://factordb.com/index.php?showid=1100000002383583629&base=16"]32235[/URL]
17 F7(0^186767)1 262*17^186768+1 [URL="http://factordb.com/index.php?showid=1100000000765961429&base=17"]186770[/URL]
18 8(0^298)B 8*18^299+11 [URL="http://factordb.com/index.php?showid=1100000002355574745&base=18"]300[/URL]
19 FG(6^110984) (904*19^110984-1)/3 [URL="http://factordb.com/index.php?showid=1100000000808118212&base=19"]110986[/URL]
20 C(D^2449) (241*20^2449-13)/19 [URL="http://factordb.com/index.php?showid=1100000002325393915&base=20"]2450[/URL]
21 C(F^479147)0K (51*21^479149-1243)/4 [URL="http://factordb.com/index.php?showid=1100000000805209046&base=21"]479150[/URL]
22 K(0^760)EC1 20*22^763+7041 [URL="http://factordb.com/index.php?showid=1100000000632724415&base=22"]764[/URL]
23 9(E^800873) (106*23^800873-7)/11 [URL="http://factordb.com/index.php?showid=1100000000782858648&base=23"]800874[/URL]
24 2(0^313)7 2*24^314+7 [URL="http://factordb.com/index.php?showid=1100000002355610241&base=24"]315[/URL]
25 9(6^136965)M (37*25^136966+63)/4 [URL="http://factordb.com/index.php?showid=1100000000808118185&base=25"]136967[/URL]
26 (M^8772)P (22*26^8773+53)/25 [URL="http://factordb.com/index.php?showid=1100000000758011195&base=26"]8773[/URL]
27 A(0^109003)PM 10*27^109005+697 [URL="http://factordb.com/index.php?showid=1100000000808118203&base=27"]109006[/URL]
28 O4(O^94535)9 (6092*28^94536-143)/9 [URL="http://factordb.com/index.php?showid=1100000000808118231&base=28"]94538[/URL]
29 O(0^174236)FPL 24*29^174239+13361 [URL="http://factordb.com/index.php?showid=1100000000808118178&base=29"]174240[/URL]
30 O(T^34205) 25*30^34205-1 [URL="http://factordb.com/index.php?showid=1100000000800812865&base=30"]34206[/URL]
31 IE(L^29787) (5727*31^29787-7)/10 [URL="http://factordb.com/index.php?showid=1100000002621742375&base=31"]29789[/URL]
32 S(U^9748)L (898*32^9749-309)/31 [URL="http://factordb.com/index.php?showid=1100000001550077250&base=32"]9750[/URL]
33 N7(0^610411)1 766*33^610412+1 [URL="http://factordb.com/index.php?showid=1100000000838755581&base=33"]610414[/URL]
34 US(0^9374)R 1048*34^9375+27 [URL="http://factordb.com/index.php?showid=1100000001550091394&base=34"]9377[/URL]
35 1B(0^56061)1 46*35^56062+1 [URL="http://factordb.com/index.php?showid=1100000000885460611&base=35"]56064[/URL]
36 (P^81993)SZ (5*36^81995+821)/7 [URL="http://factordb.com/index.php?showid=1100000002394962083&base=36"]81995[/URL]
[/CODE]

(in sequences below, 0 means no such prime exists, [I]Italic type[/I] means either not minimal prime (start with b+1) in base b or not acceptable as the form will produce a digit >=b or <0 in base b)

Length of the smallest repunit prime (form: {1}) in base b for b = 2, 3, 4, ..., 160:

2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, 0, 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2, 3, 2, 19, 97, 3, 2, 17, 2, 17, 3, 3, 2, 23, 29, 7, 59, 3, 5, 3, 5, 0, 5, 43, 599, 0, 2, 5, 7, 5, 2, 3, 47, 13, 5, 1171, 2, 11, 2, 163, 79, 3, 1231, 3, 0, 5, 7, 3, 2, 7, 2, 13, 270217, 3, 5, 3, 2, 17, 7, 13, 7

Length of the smallest generalized Fermat prime (form: 1{0}1) in base b for b = 2, 3, 4, ..., 160:

2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 17, 0, 2, 0, 5, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 0, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 17, 0, 2, 0, 5, 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 2, 0, 33, 0, 3, 0, 5, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 3, 0, 2, 0, 2, 0, 5, 0, 5, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 2, 0, 9, 0, 5, 0, 2, 0, 17, 0, 3

Length of the smallest generalized half Fermat prime (form: {x}y, x = (b-1)/2, y = (b+1)/2) in base b for b = 2, 3, 4, ..., 160:

0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 0, 0, 2, 0, (>=524288 or 0), 0, 8, 0, 2, 0, (>=524288 or 0), 0, 2, 0, 16, 0, 8, 0, 2, 0, 8, 0, 2, 0, 2, 0, 8, 0, (>=524288 or 0), 0, 4, 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, 2, 0, 2, 0, 4, 0, 32, 0, (>=524288 or 0), 0, 2, 0, 4, 0, (>=524288 or 0), 0, 2, 0, 16, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 4, 0, 4, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 16, 0, 4, 0, 4, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, 0, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 4, 0, (>=524288 or 0), 0, 16, 0, 2, 0

Length of the smallest Williams prime of the 1st kind (form: x{y}, x = b-2, y = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since the digit 2-2 = 0 in base 2 cannot be leading digit)

[I]3[/I], 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 15, 2, 2, 3, 7, 2, 2, 2, 56, 13, 2, 134, 2, 21, 2, 3, 2, 2, 3, 16, 4, 2, 8, 136212, 2, 2, 8, 2, 8, 8, 2, 2, 2, 3, 2, 26, 2, 6, 4, 2, 2, 2, 2, 3, 4, 2, 2, 900, 4, 12, 2, 2, 2, 64, 2, 14, 2, 26, 9, 4, 3, 8, 2, 45, 3, 12, 4, 82, 21496, 2, 3, 2, 2, 4, 26, 2, 520, 78, 477, 2, 2, 3, 2, 4984, 3, 3, 2, 2, 4, 2, 4, 3, 38, 411, 7, 6, 3, 8, 286644, 3, 2, 2, 3, 3, 4, 3, 2, 4, 7, 34, 8740, 2, 2, (>2220000 or 0), 3, 9, 2, 2, 3, 4, 2, 6, 26, 3, 2, 24, 2, 2, 8, 3, 2, 2, 6, 4, 2, 2, 4, 4, 3, 2, 2, 2, 4, 128, 2, 2

Length of the smallest Williams prime of the 2nd kind (form: x{0}1, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 3, 2, 2, 3, 2, 4, 11, 4, 2, 3, 2, 2, 5, 2, 30, 15, 2, 2, 15, 3, 2, 3, 5, 2, 3, 5, 6, 13, 3, 2, 3, 3, 10, 17, 2, 3, 81, 2, 3, 5, 3, 4, 17, 3, 3, 3, 2, 16, 961, 16, 2, 5, 4, 2, 15, 2, 7, 21, 2, 4, 947, 7, 2, 19, 11, 2, 5, 2, 6, 43, 5, 2, 829, 2, 2, 3, 2, 13, 3, 7, 5, 31, 4, 3023, 3, 2, 2, 9, 3, 5, 5, 3, 12, 9, 3, 2, 3, 2, 57, 3, 13, 2, 5, 6, 16, 3, 2, 2, 5, 4, 3, 17, 4, 2, 47, 2, 3, 6217, (>400000 or 0), 3, 17, 5, 166, 73, 6, 65, 15, 2, 3, 51, 3, 280, 13, 3, 2, 3, 7, 2, 5, 2, 4, 5, 5, 2, 3, 15, 2, 9, 5, 2, 7, 2, 30, 1621, 17, 6

Length of the smallest Williams prime of the 4th kind (form: 11{0}1) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1)

[I]3[/I], 3, 0, 3, [I]3[/I], 0, 3, 4, 0, 4, [I]3[/I], 0, 3, 3, 0, 3, [I]11[/I], 0, 3, 3, 0, 4, 3, 0, [I]4[/I], 3, 0, 7, [I]4[/I], 0, 7, 3, 0, 4, [I]5[/I], 0, 3, 5, 0, 3, [I]4[/I], 0, 4, 4, 0, 4, [I]8[/I], 0, 3, 185, 0, 4, 3, 0, [I]4[/I], 3, 0, 3, [I]23[/I], 0, 3, 187, 0, 5, [I]3[/I], 0, 4, 3, 0, 3, [I]122[/I], 0, [I]4[/I], 3, 0, 3, [I]3[/I], 0, 3, 10, 0, 7, [I]11[/I], 0, 4, 4, 0, 3, 3, 0, 4, 5, 0, 11, [I]16[/I], 0, 5, 3, 0, 3, [I]7[/I], 0, 7, 3, 0, 82, [I]400[/I], 0, 3, 3, 0, 5, 5, 0, [I]46[/I], 3, 0, 3, [I]4[/I], 0, 4, 4, 0, 4, [I]7[/I], 0, 5, 56, 0, 3, [I]56[/I], 0, [I]11[/I], 19, 0, 22, [I]3[/I], 0, 4, 3, 0, 3, 5, 0, [I]5[/I], 3, 0, 11, [I]3[/I], 0, 4, 3, 0, 3, [I]5[/I], 0, [I]143[/I], 34, 0

Length of the smallest dual Williams prime of the 1st kind (form: {x}1, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3, (>60000 or 0), 4, 3, 7, 6, 5, 3, 2, 3, 2, 5, 55, 4, 2, 7, 4, 4, 61, 2, 2, (>25000 or 0), 991, 4, 3, 18, 2, 9, 2, 4, 61, 17, 9, 3, 16, 18, 401, 3, 3, 25, 2, 9, 3, 13, 3, 5, 4, 2, 3, 3, 2, 281, 2, 255, 5, 3, 2, 7, 90, 2, (>25000 or 0), 6, 2, 3, 2, 6, (>25000 or 0), 6, 33

Length of the smallest dual Williams prime of the 2nd kind (form: 1{0}x, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 17, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 4, 3, 2, 3, 11, 2, 2, 109, 4, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 3, 21, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 3, 8, 9, 4, 2, 3, 2, 25, 3, 2, 2, 13, 5, 4, 9, 2, 2, 5, 4, 2, 195, 4, 2, 3, 2, 3, 3, 2, 9, 3, 2, 2, 5, 3, 3, 55, 2, 2, 5, 2, 2, 3, 45, 3, 15, 4, 2, 1401, 7, 4, 5, 7, 2, 20089, 2, 2, 7, 2, 7, 5, 2, 2, 5, 64371, 4, 3, 2, 4, 505, 2, 3, 3, 2, 9, 3, 2, 2, 61, 3, 2, 3, 2, 2, 5, 3, 3, 9, 2, 3, 3, 4, 4, 21, 3, 2, 3, 2, 2, 3, 2, 3

Length of the smallest dual Williams prime of the 4th kind (form: 1{0}11) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1)

2, 2, 0, 2, 2, 0, 2, 2, 0, 2, [I]3[/I], 0, 2, 2, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 3, 2, 0, 2, [I]4[/I], 0, 2, 7, 0, 5, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 3, [I]4[/I], 0, 3, 2, 0, 2, [I]3[/I], 0, 2, 2, 0, 3, [I]5[/I], 0, 2, 2, 0, 3, 2, 0, 3, 2, 0, 2, [I]31[/I], 0, 2, 4, 0, 2, 2, 0, 8, 68, 0, 2, 2, 0, 2, 2, 0, 3, [I]4[/I], 0, 4, 2, 0, 5, [I]4[/I], 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 13, 8, 0, 2, [I]4[/I], 0, 2, 569, 0, 2, [I]25[/I], 0, 2, 2, 0, 44, 2, 0, 2, 2, 0, 3, 4, 0, 2, 3, 0, 8, [I]3[/I], 0, 4, 2, 0, 2, 2, 0, 2, 5, 0

Length of the smallest prime of the form 2*b^n+1 (form: 2{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2)

[I]2[/I], 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 2, 2, 0, 48, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 2730, 2, 0, 2, 3, 0, 2, 3, 0, 176, 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 4, 4, 0, 44, 2, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 4, 2, 0, 12, 2, 0, 2, 5, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 2, 2, 0, 192276, 3, 0, 1234, 2, 0, 4, 6, 0, 52, 2, 0, 2, 2, 0, 2, 287, 0, 2, 2, 0, 756, 3, 0, 2, 5, 0, 2, 7, 0, 2, 3, 0, 2, 2, 0, 328, 2, 0, 2, 2, 0, 6, 6, 0, 2, 155, 0, 4, 4, 0, 4, 2, 0, 2, 2, 0, 2, 4, 0

Length of the smallest prime of the form 2*b^n-1 (form: 1{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 3, 5, 2, 2, 3, 3, 2, 11, 2, 2, 7, 2, 3, 7, 2, 3, 137, 2, 2, 7, 7, 2, 7, 2, 2, 3, 3, 2, 3, 2, 3, 5, 2, 3, 5, 5, 2, 3, 2, 2, 45, 2, 2, 3, 2, 4, 3, 6, 4, 3, 3, 2, 5, 2, 769, 5, 2, 2, 53, 35, 3, 133, 2, 2, 15, 8, 2, 3, 3, 2, 9, 2, 3, 11, 2, 25, 61, 2, 2, 3, 4, 6, 3, 2, 2, 3, 2, 2, 43, 3, 5, 69, 7, 2, 21911, 3, 3, 17, 25, 2, 3, 2, 2, 33, 2, 3, 29, 2, 2, 7, 9, 5, 3, 2, 3, 19, 2, 4, 5, 2, 5, 3, 2, 2, 3, 5, 2, 3, 2, 2, 3, 25, 13, 17, 2, 5, 5, 9, 6, 797, 3, 2, 3, 2, 2, 3, 2, 3

Length of the smallest prime of the form b^n+2 (form: 1{0}2) in base b for b = 2, 3, 4, ..., 160:

0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 12, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 114, 0, 0, 0, 2, 0, 8, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 13, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 256, 0, 0, 0, 9, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 16, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 24, 0, 0, 0, 2, 0, 2, 0, 0, 0, 5, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 137, 0

Length of the smallest prime of the form b^n-2 (form: {x}y, x = b-1, y = b-2) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle, also (b,n) = (2,2) is also not acceptable, although 2^2-2 is prime, since 2^2-2 is not a prime which is >2, but this puzzle requires primes >b)

0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 6, 0, 2, 0, 2, 0, 24, 0, 7, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 2, 0, 11, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 12, 0, 4, 0, 2, 0, 2, 0, 8, 0, 3, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 38, 0, 130, 0, 4, 0, 4, 0, 4, 0, 2, 0, 3, 0, 2, 0, 4, 0, 747, 0, 3, 0, 4, 0, 2, 0, 10, 0, 2, 0, 3, 0, 17, 0, 10, 0, 13, 0, 2, 0, 2, 0, 2, 0, 6, 0, 42, 0, 2, 0, 3, 0, 2, 0, 6, 0, 2, 0, 10, 0, 2, 0, 4, 0, 4, 0, 2, 0, 16, 0, 50, 0, 3, 0, 9, 0, 2, 0, 22, 0, 25, 0

Length of the smallest prime of the form 3*b^n+1 (form: 3{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2)

[I]2[/I], 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 6, 0, 3, 0, 2, 0, 13, 0, 2, 0, 2, 0, 3, 0, 2, 0, 15, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 5, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 6, 0, 2, 0, 2, 0, 4, 0, 271, 0, 2, 0, 2, 0, 13, 0, 2, 0, 47, 0, 3, 0, 2, 0, 2, 0, 2, 0, 28, 0, 22, 0, 2, 0, 5, 0, 2, 0, 9, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 3, 0, 5, 0, 3

Length of the smallest prime of the form 3*b^n-1 (form: 2{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2)

2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 2, 0, 12, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2524, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 60, 0, 2, 0, 2, 0, 11, 0, 3, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 15, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 7, 0, 3, 0, 51, 0, 64, 0, 2, 0, 2, 0, 2, 0, 3, 0, 12, 0, 51, 0, 2, 0, 2, 0, 39, 0, 2, 0, 3, 0, 3, 0, 2, 0, 27, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 0, 2

Length of the smallest prime of the form b^n+3 (form: 1{0}3) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2)

[I]2[/I], 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 21, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 2, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 16, 0, 2, 0, 0, 0, 3, 0, 2

Length of the smallest prime of the form b^n-3 (form: {x}y, x = b-1, y = b-3) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=2 case is not acceptable, since there is no digit 2-3 = -1 in base 2)

[I]3[/I], 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 6, 0, 0, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 21, 0, 105, 0, 0, 0, 18, 0, 2, 0, 0, 0, 5, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 3, 0, 5, 0, 0, 0, 2, 0, 13, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 204, 0, 0, 0, 2, 0, 70, 0, 0, 0, 4, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 6, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 7, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 346, 0, 396, 0, 0, 0, 3, 0, 21

Length of the smallest prime of the form 4*b^n+1 (form: 4{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3, 4 cases are not acceptable, since there is no digit 4 in bases 2, 3, 4)

[I]3[/I], [I]2[/I], [I]2[/I], 3, 0, 2, 3, 2, 2, 0, 3, 2, 0, 2, 0, 7, 2, 4, 3, 0, 2, 343, 2, 2, 0, 2, 2, 0, 7, 0, (>=1717986919 or 0), 3, 2, 43, 0, 2, 11, 2, 6, 0, 3, 2, 0, 2, 0, 3, 2, 2, 11, 0, 11, (>1670000 or 0), 4, 3, 0, 2, 2, 0, 2, 0, 3, 3, 2, 3, 0, 2, 7, 2, 2, 0, 4, 2, 0, 3, 0, 6099, 2, 2, 3, 0, 7, 5871, 2, 3, 0, 2, 2, 0, 3, 0, 3, 2, 4, 7, 0, 2, 295, 2, 2, 0, 2, 3, 0, 2, 0, 32587, 2, 4, 11, 0, 2, 2959, 2, 2, 0, 102, 3, 0, 3, 0, 359, 7, 472, 3, 0, 2, 3, 20, 2, 0, 3, 6, 0, 2, 0, 19, 4, 2, 3, 0, 2, 11, 2, 22, 0, 4, 2, 0, 2, 0, 19, 2, 2, (>1280000 or 0), 0, 3, 875, 30, 2

Length of the smallest prime of the form 4*b^n-1 (form: 3{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3 cases are not acceptable, since there is no digit 3 in base 2, 3)

[I]2[/I], [I]2[/I], 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 6, 0, 0, 2, 2, 0, 0, 4, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1556, 2, 0, 2, 4, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 10, 2, 0, 10, 2, 0, 2, 0, 0, 2, 1119850, 0, 0, 6, 0, 2, 2, 0, 8, 0, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 42, 0, 2, 2, 0, 4, 0, 0, 4, 14, 0, 0, 2, 0, 252, 2, 0, 2, 2, 0, 4, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 16, 0, 0, 2, 4, 0, 0, 6, 0, 2, 14, 0, 6, 2, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 2, 0, 0

Length of the smallest prime of the form b^n+4 (form: 1{0}4) in base b for b = 2, 3, 4, ..., 160: (the b=3 case is not acceptable, since there is no digit 4 in base 3)

0, [I]2[/I], 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 3, 0, 2, 0, 0, 0, 13403, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 47, 0, 2, 0, 0, 0, 83, 0, 2, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 7, 0, 2, 0, 0, 0, 10647, 0, 3, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 4, 0, 0, 0, 2, 0, 2, 0, 3, 0, (>25000 or 0), 0, 0, 0, 71, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 214, 0, 2, 0

Length of the smallest prime of the form b^n-4 (form: {x}y, x = b-1, y = b-4) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=3 case is not acceptable, since there is no digit 3-4 = -1 in base 3)

0, [I]2[/I], 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 13, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 65, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 0, 0, 45, 0, 0, 0, 13, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 29, 0, 0, 0, 0, 0, 105, 0, 45, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 7, 0, 0, 0, 13, 0, 13, 0, 0, 0, 3, 0, 0, 0, 0, 0, 299, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 165, 0, 147, 0, 0, 0, 395, 0, 23, 0, 0, 0, 3, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0, 0, 0, 0

sweety439 2021-01-14 04:55

Large minimal prime (start with b+1) for bases 17<=b<=64 not in the [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]list for bases 2 to 30[/URL] or [URL="https://github.com/RaymondDevillers/primes"]list for bases 28 to 50[/URL] (because they contain single-digit primes, or because they are too large (length > 10000), primes with the latter case but not the former case are already minimal even if single-digit primes are included, and they are marked by "**") given by: (using A−Z to represent digit values 10 to 35, a−z to represent digit values 36 to 61, # to represent digit value 62, $ to represent digit value 63)

Base 17: F70[SUB]186767[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17"]CRUS generalized Sierpinski conjecture base 17[/URL])
Base 17: 970[SUB]166047[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17"]CRUS generalized Sierpinski conjecture base 17[/URL])
Base 17: 570[SUB]51310[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17"]CRUS generalized Sierpinski conjecture base 17[/URL])
Base 17: 530[SUB]4867[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17"]CRUS generalized Sierpinski conjecture base 17[/URL])
Base 17: 10[SUB]9019[/SUB]1F (found by [URL="http://www.primenumbers.net/prptop/searchform.php?form=17%5En%2B32&action=Search"]Guido Smetrijns[/URL])
Base 18: 80[SUB]298[/SUB]B
Base 19: F10[SUB]18523[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19"]CRUS generalized Sierpinski conjecture base 19[/URL])
Base 20: CD[SUB]2449[/SUB] (found by me)
Base 21: 5D0[SUB]19848[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S21"]CRUS generalized Sierpinski conjecture base 21[/URL])
Base 24: 20[SUB]313[/SUB]7 (note: F1[SUB]957[/SUB] is not minimal prime (start with b+1), since its repeating digit is 1)
Base 27: JD0[SUB]7667[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S27"]CRUS generalized Sierpinski conjecture base 27[/URL])
Base 29: 10[SUB]8095[/SUB]A (found by [URL="http://www.primenumbers.net/prptop/searchform.php?form=29%5En%2B10&action=Search"]Ray Chandler[/URL])
Base 30: OT[SUB]34205[/SUB] (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R30"]CRUS generalized Riesel conjecture base 30[/URL])
**Base 31: E8U[SUB]21866[/SUB]P (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: IEL[SUB]29787[/SUB] (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: LF[SUB]21052[/SUB]G (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: MIO[SUB]10747[/SUB]L (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: PEO0[SUB]22367[/SUB]Q (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: L[SUB]10012[/SUB]9G (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 31: R[SUB]22137[/SUB]1R (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
Base 33: 130[SUB]23614[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33"]CRUS generalized Sierpinski conjecture base 33[/URL])
Base 33: N70[SUB]610411[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33"]CRUS generalized Sierpinski conjecture base 33[/URL])
**Base 36: P[SUB]81993[/SUB]SZ (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
Base 37: 1F0[SUB]1627[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
Base 37: 910[SUB]6840[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
**Base 37: FYa[SUB]22021[/SUB] (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R37"]CRUS generalized Riesel conjecture base 37[/URL])
Base 37: HZ0[SUB]2148[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
Base 37: PB0[SUB]8607[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
**Base 37: R8a[SUB]20895[/SUB] (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R37"]CRUS generalized Riesel conjecture base 37[/URL])
Base 37: Z10[SUB]6195[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
Base 38: 20[SUB]2728[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S38"]CRUS generalized Sierpinski conjecture base 38[/URL])
Base 38: Lb[SUB]1579[/SUB] (found by me)
Base 38: V0[SUB]1527[/SUB]1 (found by me)
Base 38: ab[SUB]136211[/SUB] (found by [URL="https://harvey563.tripod.com/wills.txt"]Williams primes search[/URL])
**Base 40: QaU[SUB]12380[/SUB]X (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
Base 42: 2f[SUB]2523[/SUB] (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R42"]CRUS generalized Riesel conjecture base 42[/URL])
Base 43: F30[SUB]194122[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S43"]CRUS generalized Sierpinski conjecture base 43[/URL])
Base 45: Ni[SUB]153355[/SUB] (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R45"]CRUS generalized Riesel conjecture base 45[/URL])
**Base 45: O0[SUB]18521[/SUB]1 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S45"]CRUS generalized Sierpinski conjecture base 45[/URL])
Base 48: T0[SUB]133041[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S48"]CRUS generalized Sierpinski conjecture base 48[/URL])
**Base 49: 11c0[SUB]29736[/SUB]1 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S49"]CRUS generalized Sierpinski conjecture base 49[/URL])
**Base 49: Fd0[SUB]18340[/SUB]1 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S49"]CRUS generalized Sierpinski conjecture base 49[/URL])
**Base 49: SLm[SUB]52698[/SUB] (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R49"]CRUS generalized Riesel conjecture base 49[/URL])
**Base 49: Ydm[SUB]16337[/SUB] (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm#R49"]CRUS generalized Riesel conjecture base 49[/URL])
Base 50: 70[SUB]515[/SUB]1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S50"]CRUS generalized Sierpinski conjecture base 50[/URL])
**Base 51: 1[SUB]4229[/SUB] (already minimal even if single-digit primes are included) (found by [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]generalized repunit prime search[/URL])
**Base 51: c0[SUB]4880[/SUB]1 (already minimal even if single-digit primes are included) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S51"]CRUS generalized Sierpinski conjecture base 51[/URL])
**Base 52: g0[SUB]4821[/SUB]1 (already minimal even if single-digit primes are included) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S52"]CRUS generalized Sierpinski conjecture base 52[/URL])
Base 53: 10[SUB]13401[/SUB]4 (found by me)
**Base 53: 80[SUB]227182[/SUB]1 (already minimal even if single-digit primes are included) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S53"]CRUS generalized Sierpinski conjecture base 53[/URL])
**Base 57: E0[SUB]14954[/SUB]1 (already minimal even if single-digit primes are included) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S57"]CRUS generalized Sierpinski conjecture base 57[/URL])
**Base 58: L0[SUB]1030[/SUB]1 (already minimal even if single-digit primes are included) (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S58"]CRUS generalized Sierpinski conjecture base 58[/URL])
**Base 60: L0[SUB]289[/SUB]mmn (already minimal even if single-digit primes are included) (found by me)
**Base 60: Q[SUB]896[/SUB]1 (already minimal even if single-digit primes are included) (found by me)
**Base 60: e[SUB]1937[/SUB]1 (already minimal even if single-digit primes are included) (found by me)
**Base 60: g[SUB]786[/SUB]Un (already minimal even if single-digit primes are included) (found by me)
**Base 60: n[SUB]437[/SUB]Fn (already minimal even if single-digit primes are included) (found by me)
Base 64: N$[SUB]3020[/SUB] (found by [URL="http://www.prothsearch.com/riesel2.html"]Riesel prime search[/URL])

For more minimal primes (start with b+1) for bases b>16, see [URL="https://github.com/xayahrainie4793/non-single-digit-primes/blob/main/smallest%20generalized%20near-repdigit%20prime.txt"]x{y} and {x}y[/URL] [URL="https://github.com/xayahrainie4793/non-single-digit-primes/blob/main/smallest%20prime%20of%20the%20form%20x000000y.txt"]x{0}y[/URL] [URL="https://github.com/xayahrainie4793/non-single-digit-primes/blob/main/x0000yz%20and%20xy0000z"]x{0}yz and xy{0}z[/URL]

Also see post [URL="https://mersenneforum.org/showpost.php?p=560547&postcount=43"]https://mersenneforum.org/showpost.php?p=560547&postcount=43[/URL]

Some known unsolved families for bases b<=64 not in the [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]list for bases 2 to 30[/URL] or [URL="https://github.com/RaymondDevillers/primes"]list for bases 28 to 50[/URL]:

Base 11: 5{7} (found by me)
Base 13: 9{5} (found by me)
Base 13: A{3}A (found by me)
Base 16: {3}AF (found by me)
Base 16: {4}DD (found by me)
Base 17: 15{0}D (found by me)
Base 17: 1F{0}7 (found by me)
Base 18: C{0}C5 (found by me)
Base 23: H3{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529"]CRUS generalized Sierpinski conjecture base 529[/URL])
Base 23: JH{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529"]CRUS generalized Sierpinski conjecture base 529[/URL])
Base 25: F{2} (found by extended generalized Riesel conjecture base 25 with k > CK)
Base 31: 2{F} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 31[/URL])
Base 31: 3{5} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 31[/URL])
Base 32: S{V} (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024"]CRUS generalized Riesel conjecture base 1024[/URL]) (note: S(V^1745576) (base 32) is 3-PRP but not prime, see [URL="https://primes.utm.edu/primes/page.php?id=122375&deleted=1"]https://primes.utm.edu/primes/page.php?id=122375&deleted=1[/URL])
Base 37: 2K{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
Base 37: {I}J (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])
Base 38: 1{0}V (see [URL="https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31"]https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31[/URL])
Base 43: 2{7} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 43[/URL])
Base 43: 3b{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S43"]CRUS generalized Sierpinski conjecture base 43[/URL])
Base 53: 19{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S53"]CRUS generalized Sierpinski conjecture base 53[/URL])
Base 53: 4{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S53"]CRUS generalized Sierpinski conjecture base 53[/URL])
Base 55: a{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S55"]CRUS generalized Sierpinski conjecture base 55[/URL])
Base 55: {R}S (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])
Base 60: Z{x} (see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base60-reserve.htm"]CRUS generalized Riesel conjecture base 60[/URL])
Base 62: 1{0}1 (see [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL])
Base 63: {V}W (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])

sweety439 2021-01-14 19:17

For more lengths for the smallest prime of the form k*b^n+1 (form: k{0}1) or k*b^n-1 (form: x{y}, x = k-1, y = b-1) with k<b in base b, see [URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL]

Note: The requiring of this project (to solve this puzzle) is that [B]k<b[/B], but the requiring of CRUS is that [B]k<CK[/B], since the CK value may be either >b or <b, thus this is neither necessary nor sufficient, but they have many intersection (when k<min(b,CK)), e.g. 8*23^n+1 (8{0}1 in base 23), 25*30^n-1 (O{T} in base 30), 2*38^n+1 (2{0}1 in base 38), 3*42^n-1 (2{f} in base 42), 36*48^n+1 (a{0}1 in base 48), 4*53^n+1 (4{0}1 in base 53), etc., they corresponding to the minimal primes (start with b+1) 80[SUB]119214[/SUB]1 in base 23 (8*23^119215+1), OT[SUB]34205[/SUB] in base 30 (25*30^34205-1), 20[SUB]2728[/SUB]1 in base 38 (2*38^2729+1), 2f[SUB]2523[/SUB] in base 42 (3*42^2523-1), unsolved family a{0}1 in base 48 searched to length 500001 (36*48^n+1 searched to n=500000), unsolved family 4{0}1 in base 53 searched to length 1700001 (4*53^n+1 searched to n=1700000), etc.

Also this project (to solve this puzzle) includes unsolved family 4{0}1 in base 32 searched to length (2^33-2)/5 = 1717986918 (since all primes of the form 4{0}1 in base 32 must be Fermat primes, and none of the known Fermat primes (F0 to F4) are of the form 4{0}1 in base 32 (their base 32 forms are 3, 5, H, 81, 2001), and all Fermat numbers F5 to F32 are known to be composite, see [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL], thus, the smallest possible prime of the form 4{0}1 in base 32 is F33 = 40[SUB]1717986917[/SUB]1 in base 32, which has length 1717986919), which is excluded in CRUS, since CRUS excluded k's that make GFNs, i.e. q^m*b^n+1 where b is the base, m>=0, and q is a root of the base, and 4*32^n+1 = (2^2)*32^n+1, and 2 is a root of 32 (32^(1/5)).

References: [URL="https://mersenneforum.org/showpost.php?p=116415&postcount=1"]https://mersenneforum.org/showpost.php?p=116415&postcount=1[/URL] [URL="https://mersenneforum.org/showthread.php?t=6916"]https://mersenneforum.org/showthread.php?t=6916[/URL] [URL="https://mersenneforum.org/showthread.php?t=9479"]https://mersenneforum.org/showthread.php?t=9479[/URL] [URL="https://mersenneforum.org/showthread.php?p=447689#post447689"]https://mersenneforum.org/showthread.php?p=447689#post447689[/URL] [URL="https://mersenneforum.org/showthread.php?p=447998#post447998"]https://mersenneforum.org/showthread.php?p=447998#post447998[/URL] [URL="https://www.utm.edu/staff/caldwell/preprints/2to100.pdf"]https://www.utm.edu/staff/caldwell/preprints/2to100.pdf[/URL] [URL="https://oeis.org/A171381"]https://oeis.org/A171381[/URL] [URL="https://oeis.org/A182331"]https://oeis.org/A182331[/URL] [URL="https://oeis.org/A078680"]https://oeis.org/A078680[/URL] [URL="https://mersenneforum.org/showthread.php?p=586113#post586113"]https://mersenneforum.org/showthread.php?p=586113#post586113[/URL] [URL="https://www.primepuzzles.net/conjectures/conj_004.htm"]https://www.primepuzzles.net/conjectures/conj_004.htm[/URL] [URL="https://arxiv.org/pdf/1605.01371.pdf"]https://arxiv.org/pdf/1605.01371.pdf[/URL] (this reference shows that the property of the existence of a Fermat prime > F4 is at most 10^(-9), and thus base 32 (also bases 128, 512, 1024) is virtually impossible to solve with current knowledge and technology, for the similar problem to other bases, it is excepted that the number of primes of the form b^(2^n)+1 (for fixed even base b) or (b^(2^n)+1)/2 (for fixed odd base b) is finite (such forms are called GFN (generalized Fermat numbers, i.e. b^(2^n)+1 (for even base b)) or half GFN (generalized half Fermat numbers, i.e. (b^(2^n)+1)/2 (for odd base b)), and the families which all possible primes are GFN or half GFN are called GFN families or half GFN families, see [URL="https://mersenneforum.org/showthread.php?t=20427"]https://mersenneforum.org/showthread.php?t=20427[/URL]), but this is undecidable at this point in time (see page 3 of [URL="https://www.utm.edu/staff/caldwell/preprints/2to100.pdf"]https://www.utm.edu/staff/caldwell/preprints/2to100.pdf[/URL]).

For an unsolved family (i.e. families which have not yet yielded a prime, nor can it be ruled out as contain no primes > b), we except that there must be a prime at some point in this family if this family is neither GFN nor half GFN, but we except that this family contain no primes if this family is GFN or half GFN.

GFN primes can be easily proven to be prime using the N-1 method (since GFN-1 is a power of this base, thus is trivially 100% factored), but half GFN primes cannot (references: [URL="https://oeis.org/A275530"]https://oeis.org/A275530[/URL] (Batalov's comment: The terms of this sequence with n > 11 correspond to probable primes which are too large to be proven prime currently) [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL] (the factorization of N-1 for the PRP (71^16384+1)/2) [URL="https://www.naturalspublishing.com/files/published/icud9179k95t58.pdf"]https://www.naturalspublishing.com/files/published/icud9179k95t58.pdf[/URL]), since for the half GFN primes, take these N-1 or N+1, there is nothing immediately [URL="https://en.wikipedia.org/wiki/Smooth_number"]smooth[/URL] about them (like that the [URL="https://www.rieselprime.de/ziki/Carol-Kynea_prime"]Carol/Kynea primes[/URL] can be easily proven to be prime using the N-1 method, but the [URL="https://mersenneforum.org/showthread.php?t=25409"]half Carol/Kynea primes[/URL] cannot, since for the half Carol/Kynea primes, take these N-1 or N+1, there is nothing immediately [URL="https://en.wikipedia.org/wiki/Smooth_number"]smooth[/URL] about them, see [URL="https://mersenneforum.org/showpost.php?p=541285&postcount=4"]this post[/URL]), even neither N-1 nor N+1 can be easily factored with factored part >= 33.3333%, and thus there are many half GFN PRPs which are not proven to be primes (reference: [URL="https://primes.utm.edu/primes/search.php?Comment=generalized%20Fermat&Number=10000"]top GFN primes[/URL] and [URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En%2B1%29%2F2&action=Search"]top half GFN PRPs[/URL]), however, [I]all[/I] known first half GFN primes in bases 2<=b<=1024 are proven primes, the largest such prime is [URL="http://factordb.com/index.php?id=1100000000094956024"](827^1024+1)/2[/URL], although there are half GFN PRPs in bases 2<=b<=1024 which have not proven to be primes, such as [URL="http://factordb.com/index.php?id=1100000000213085670"](71^16384+1)/2[/URL] and [URL="http://factordb.com/index.php?id=1100000000710475118"](799^2048+1)/2[/URL] (edit: (799^2048+1)/2 is now proven prime, its [URL="http://www.ellipsa.eu/public/primo/primo.html"]PRIMO[/URL] primality certificate is [URL="http://factordb.com/cert.php?id=1100000000710475118"]http://factordb.com/cert.php?id=1100000000710475118[/URL]), but they are not first half GFN primes to corresponding bases, the first half GFN primes to these two bases are (71^2+1)/2 and (799^2+1)/2, respectively, besides, for examples of largest GFN primes in bases 2<=b<=1024, see [URL="http://factordb.com/index.php?id=1100000000090082284"]150^2048+1[/URL] and [URL="http://factordb.com/index.php?id=1100000000094955949"]824^1024+1[/URL], since for such primes N-1 are trivially 100% factored, they can be easily proven to be prime using the N-1 method.

For GRU primes, only Mersenne primes can easily proven to be prime (using the N+1 method, since Mersenne prime + 1 is a power of 2, thus is trivially 100% factored), since for GRU primes with base b>2, neither N-1 nor N+1 can be easily factored with factored part >= 33.3333%, and thus there are many half GRU PRPs which are not proven to be primes (reference: [URL="https://primes.utm.edu/primes/search.php?Comment=generalized%20repunit&Number=10000"]top GRU primes[/URL] and [URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-1%29%2Fd&action=Search"]top GRU PRPs[/URL])

(for the list of known GFN primes, see: [URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]sorted by n[/URL] [URL="http://jeppesn.dk/generalized-fermat.html"]sorted by base[/URL] [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]sorted by base[/URL], and for the list of known half GFN (probable) primes, see: [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]sorted by n[/URL], also see OEIS sequence [URL="https://oeis.org/A253242"]A253242[/URL] and [URL="https://oeis.org/A253242/a253242.txt"]its a-file[/URL] for the known GFN primes and half GFN (probable) primes, also see OEIS sequences [URL="https://oeis.org/A056993"]A056993[/URL] and [URL="https://oeis.org/A275530"]A275530[/URL] for the smallest GFN prime and half GFN (probable) prime for given exponent)

(for the list of known GRU primes, see: [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]sorted by base[/URL] [URL="https://archive.ph/tf7jx"]sorted by base[/URL], also see OEIS sequence [URL="https://oeis.org/A084740"]A084740[/URL] and [URL="https://oeis.org/A084740/a084740_2.txt"]its b-file[/URL] for the known GRU (probable) primes, also see OEIS sequence [URL="https://oeis.org/A066180"]A066180[/URL] for the smallest GRU (probable) prime for given exponent)

(also see: [URL="https://oeis.org/A121326"]primes which are GFN primes for some base[/URL] [URL="https://oeis.org/A027862"]primes which are half GFN primes for some base[/URL] [URL="https://oeis.org/A085104"]primes which are GRU primes for some base[/URL])

Note: See article [URL="https://www.researchgate.net/profile/Mercedes-Orus-Lacort/publication/338701495_Fermat_Numbers/links/5e260d8092851c89c9b59e22/Fermat-Numbers.pdf"]https://www.researchgate.net/profile/Mercedes-Orus-Lacort/publication/338701495_Fermat_Numbers/links/5e260d8092851c89c9b59e22/Fermat-Numbers.pdf[/URL], maybe in future it can be proven that there are only 5 Fermat primes {3, 5, 17, 257, 65537}, thus the base 32 families 4{0}1 and G{0}1, etc. can be ruled out as only contain composites, however, currently it is still not proven and is an unsolved problem.

Thus, in some bases there exist families which are excepted as contain no primes, but undecidable at this point in time (they are exactly the GFN families or half GFN families in bases 2<=b<=1024 with no known (probable) primes), thus these bases are almost impossible to solve at this time. Bases 2<=b<=1024 which I am aware of with this problem are 31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024 (totally 369 such bases of the 1023 bases 2<=b<=1024, thus there are 1023-369=654 bases of the 1023 bases 2<=b<=1024 which [I]might[/I] be solved at this time)), for such even bases b, this is GFN, and for such odd bases b, this is half GFN. (I think that this may be the reason why [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL] and [URL="https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf"]https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf[/URL] stop at base 30, instead of the usual base 36 (which is the largest base that the digits can be written using numbers and English letters, i.e. the digits can be written using 0-9 and A-Z, see [URL="https://archive.ph/gmMRY"]https://archive.ph/gmMRY[/URL] and [URL="https://baseconvert.com/"]https://baseconvert.com/[/URL] and [URL="https://www.dcode.fr/base-36-cipher"]https://www.dcode.fr/base-36-cipher[/URL] and [URL="http://www.tonymarston.net/php-mysql/converter.html"]http://www.tonymarston.net/php-mysql/converter.html[/URL], and note that the digits for the numbers in these two article are also written using 0-9 and A-Z), since base 31 and base 32 are almost impossible to solve at this time, base 31 family {F}G is half GFN family with no known (probable) primes, and base 32 families 4{0}1 and G{0}1 are GFN families with no known primes)

(the smallest prime in GFN family or half GFN family for base b is always minimal prime (start with b+1) in base b, unless b is power of 3)

The families which are excepted as contain no primes, but undecidable at this point in time, for these 369 bases are: (totally 377 families)

* 4:{0}:1, 16:{0}:1 for b = 32
* 12:{62}:63 for b = 125 (Note: {62}:63 for b = 125 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 12:{62}:63 for b = 125 (if exists) must be minimal prime (start with b+1) in base b = 125)
* 16:{0}:1 for b = 128
* 36:{0}:1 for b = 216
* 24:{171}:172 for b = 343 (Note: {171}:172 for b = 343 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 24:{171}:172 for b = 343 (if exists) must be minimal prime (start with b+1) in base b = 343)
* 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512
* 10:{0}:1, 100:{0}:1 for b = 1000
* 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024
* 1:{0}:1 for other even bases b
* {((b-1)/2)}:((b+1)/2) for other odd bases b

Note: GFN families and half GFN families which can be ruled out as contain no primes by full algebra factorization such as family 1:{0}:1 in base 8, {13}:14 in base 27, 1:{0}:1 in base 32, 1:{0}:1 in base 64, are not listed here, see post [URL="https://mersenneforum.org/showpost.php?p=199373&postcount=67"]https://mersenneforum.org/showpost.php?p=199373&postcount=67[/URL]

Note: families 8:{0}:1, 32:{0}:1, 64:{0}:1 in base 128 can be ruled out as contain no primes, since if 2^n+1 is prime, then n must be power of 2, but 7*n+3, 7*n+5, 7*n+6 cannot be powers of 2, all powers of 2 are == 1, 2, 4 mod 7

Note: although family 13:{121}:122 in base 243 (it is half GFN family) has no known primes and cannot be ruled out as contain no primes, this family is not listed here, since the smallest prime in this family will not be minimal prime (start with b+1) in base b=243, since 13:121:121 is prime in base 243 (also, base 243 is quite strange, as this base has unsolved family which is [I]GRU[/I]: 40:{121}, this family has been searched to length >440000 with no prime or PRP found, since [URL="https://oeis.org/A028491"]A028491[/URL] has been searched to n>2200000 with no n == 4 mod 5 found, this also makes the prime 40:{121^11}:122 (which equals the largest known generalized half Fermat prime base 3: (3^64+1)/2) minimal prime (start with b+1) in base b=243)

Also, since for GFN family b^n+1 and half GFN family (b^n+1)/2, the numbers can only be prime when n is a power of 2, it is no need to [URL="https://www.rieselprime.de/ziki/Sieving"]sieve[/URL] them using the usual [URL="https://www.rieselprime.de/ziki/Sieving_program"]sieving program[/URL] such as [I]srsieve[/I], instead, we only use [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]trial division[/URL] to find the divisors for these numbers with power-of-2 n, see posts [URL="https://mersenneforum.org/showpost.php?p=95547&postcount=63"]https://mersenneforum.org/showpost.php?p=95547&postcount=63[/URL] and [URL="https://mersenneforum.org/showpost.php?p=95792&postcount=7"]https://mersenneforum.org/showpost.php?p=95792&postcount=7[/URL] and [URL="https://mersenneforum.org/showpost.php?p=568817&postcount=116"]https://mersenneforum.org/showpost.php?p=568817&postcount=116[/URL], for references of this, see: [URL="http://www.prothsearch.com/fermat.html"]b=2[/URL] [URL="http://www.prothsearch.com/GFN03.html"]b=3[/URL] [URL="http://www.prothsearch.com/GFN05.html"]b=5[/URL] [URL="http://www.prothsearch.com/GFN06.html"]b=6[/URL] [URL="http://www.prothsearch.com/GFN07.html"]b=7[/URL] [URL="http://www.prothsearch.com/GFN10.html"]b=10[/URL] [URL="http://www.prothsearch.com/GFN11.html"]b=11[/URL] [URL="http://www.prothsearch.com/GFN12.html"]b=12[/URL]

For the GRU family (b^n-1)/(b-1), the usual sieve program will remove all n, thus we remove all composite n (and leave all prime n) and use the sieve program (without removing the n with algebra factors, only remove the n with small prime factors) remove the n such that (a*b^n+c)/gcd(a+c,b-1) has a prime factor < certain limit (e.g. 10^9 or 10^12) instead of remove the n such that (a*b^n+c)/gcd(a+c,b-1) has algebra factors, since this will remove all n-values (see [URL="https://mersenneforum.org/showpost.php?p=452132&postcount=66"]this post[/URL] and [URL="https://www.mersenneforum.org/showthread.php?t=22740"]this thread[/URL])

For the "minimal prime (start with b+1) problem in base b":

A base is [I]solved[/I] if there are no unsolved families for this base and all minimal primes (start with b+1) are proven primes.
A base is [I]weakly solved[/I] if there are no unsolved families for this base but some minimal primes (start with b+1) are only probable primes.
A base is [I]almost solved[/I] if all unsolved families for this base are GFN families or half GFN families.

e.g.

* base 31 is [I]almost solved[/I] if the only unsolved family is {F}G
* base 32 is [I]almost solved[/I] if the only two unsolved families are 4{0}1 and G{0}1
* base 37 is [I]almost solved[/I] if the only unsolved family is {I}J
* base 38 is [I]almost solved[/I] if the only unsolved family is 1{0}1
* base 50 is [I]almost solved[/I] if the only unsolved family is 1{0}1
* base 55 is [I]almost solved[/I] if the only unsolved family is {R}S
* base 62 is [I]almost solved[/I] if the only unsolved family is 1{0}1
* base 63 is [I]almost solved[/I] if the only unsolved family is {V}W
etc.

For other problems about minimal sets of given sets in given base, there are also families which are excepted as contain no numbers in corresponding sets, but not proven (like our problem about minimal sets of the primes > base in bases 2<=b<=1024, the families {F}G in base 31, 4{0}1 in base 32, G{0}1 in base 32, {I}J in base 37 (note: the prime "J" is not counted since primes must be > base), 1{0}1 in base 38, 1{0}1 in base 50, {R}S in base 55, 1{0}1 in base 62, {V}W in base 63 (note: the prime "V" is not counted since primes must be > base), etc. they are excepted as contain no primes > base, but not proven), such as 3{9}8 in base 10, 6{9}8 in base 10, nonsimple family {3,9}26 in base 10, nonsimple family {3,9}86 in base 10, they are excepted as contain no [URL="https://oeis.org/A002202"]totients[/URL], but not proven (equivalent to: 4{0}3 in base 10 contains no "totients plus 5", 7{0} in base 10 contains no "totients plus 2", {3,9}0 in base 10 contains no "totients plus 4"), reference: [URL="https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf"]https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf[/URL], this article also has the reason why these four families are excepted as contain no totients, other base also has families which are excepted as contain no totients but not proven, e.g. for base 12, families 1{0}2, 3{0}2, 5{0}2, 7{0}2, B{0}2, {1}2, {3}2, {5}2, {7}2, {B}2 are excepted as contain no totients but not proven, since in base 12 there are only few totients end with 2, for such totients see [URL="https://oeis.org/A063668"]https://oeis.org/A063668[/URL], thus the problem that whether the minimal set of "totients minus 2" in base 12 is {2, 4, 6, 8, A, 90, 1300, 3B30, 133130} is still an [URL="https://primes.utm.edu/glossary/xpage/OpenQuestion.html"]open question[/URL], see posts [URL="https://mersenneforum.org/showpost.php?p=572102&postcount=119"]https://mersenneforum.org/showpost.php?p=572102&postcount=119[/URL] and [URL="https://mersenneforum.org/showpost.php?p=572225&postcount=122"]https://mersenneforum.org/showpost.php?p=572225&postcount=122[/URL] for the minimal sets of more given sets in base 10 and base 12, many of them are extremely difficult to found, much more difficult then the minimal sets of the sets of the primes > b in bases 2<=b<=36

(like that CRUS excludes the GFNs from the conjectures to make the problems solvable for [I]all[/I] bases b, for the problem in this forum (i.e. the minimal prime (start with b+1) problem), there is a way to make the problems solvable for [I]all[/I] bases b: Excluding all primes of the form ([URL="https://oeis.org/A052410"]A052410[/URL](b)^n+1)/gcd([URL="https://oeis.org/A052410"]A052410[/URL](b)+1,2), i.e. finding all minimal "primes (start with b+1) not of the form ([URL="https://oeis.org/A052410"]A052410[/URL](b)^n+1)/gcd([URL="https://oeis.org/A052410"]A052410[/URL](b)+1,2)" in base b (note that CRUS does not include the GFN primes even if they have known primes, e.g. for base 20, CRUS does not include 1*20^2+1, and only includes 2*20^1+1, 3*20^1+1, 4*20^2+1, 5*20^1+1, 6*20^15+1, 7*20^2+1), but this exclusion is more complex and that is for a different project somewhere down the road)

sweety439 2021-02-15 16:31

Just let you know, I know the set of the minimal primes (start with b+1) <=2^32 for all bases 2<=b<=128, and I know exactly what bases 2<=b<=1024 have these families as unsolved families (at length 25K) for the minimal primes (start with b+1) problem: (also, I know exactly what bases 2<=b<=1024 where these families are ruled out as contain no primes >b)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26)

(if such forms are interpretable in the bases, e.g. "C" (means 12 (twelve)) is only interpretable in bases b>=13, and "u" (means b−6) is only interpretable in bases b>=6 (if "u" appears as the first digit, then it is only interpretable in bases b>=7, since numbers cannot have leading zeros)

* {1}
* 1{0}1
* 1{0}2
* 1{0}3
* 1{0}4
* 1{0}5
* 1{0}6
* 1{0}7
* 1{0}8
* 1{0}9
* 1{0}A
* 1{0}B
* 1{0}C
* 1{0}D
* 1{0}E
* 1{0}F
* 1{0}G
* 1{0}z
* 2{0}1
* 2{0}3
* 3{0}1
* 3{0}2
* 3{0}4
* 4{0}1
* 4{0}3
* 5{0}1
* 6{0}1
* 7{0}1
* 8{0}1
* 9{0}1
* A{0}1
* B{0}1
* C{0}1
* D{0}1
* E{0}1
* F{0}1
* G{0}1
* z{0}1
* 1{2}
* 1{3}
* 1{4}
* 1{5}
* 1{6}
* 1{7}
* 1{8}
* 1{9}
* 1{A}
* 1{B}
* 1{C}
* 1{D}
* 1{E}
* 1{F}
* 1{G}
* 1{#} (for odd base b, # = (b−1)/2)
* {2}1
* {3}1
* {4}1
* {5}1
* {6}1
* {7}1
* {8}1
* {9}1
* {A}1
* {B}1
* {C}1
* {D}1
* {E}1
* {F}1
* {G}1
* {#}1 (for odd base b, # = (b−1)/2)
* 1{z}
* 2{z}
* 3{z}
* 4{z}
* 5{z}
* 6{z}
* 7{z}
* 8{z}
* 9{z}
* A{z}
* B{z}
* C{z}
* D{z}
* E{z}
* F{z}
* y{z}
* {#}$ (for odd base b, # = (b−1)/2, $ = (b+1)/2)
* ${#} (for odd base b, # = (b−1)/2, $ = (b+1)/2)
* {y}z
* {z}1
* {z}k
* {z}l
* {z}m
* {z}n
* {z}o
* {z}p
* {z}q
* {z}r
* {z}s
* {z}t
* {z}u
* {z}v
* {z}w
* {z}x
* {z}y

Also families where the smallest prime may not be minimal prime (start with b+1):

* 1{0}11 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)
* 11{0}1 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)
* 1{0}21 (not minimal prime (start with b+1) if either 21 (2*b+1) is prime or there is smaller prime of the form 1{0}1 or 1{0}2)
* 12{0}1 (not minimal prime (start with b+1) if either 12 (b+2) is prime or there is smaller prime of the form 1{0}1 or 2{0}1)
* {1}01 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 10{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}2 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}3 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}4 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}z (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 2{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 3{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 4{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* z{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}0z (not minimal prime (start with b+1) if there is smaller prime of the form {1} or {1}z)
* 10{z} (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})
* 11{z} (not minimal prime (start with b+1) if either 11 (b+1) is prime or there is smaller prime of the form 1{z})
* {z}01 (not minimal prime (start with b+1) if there is smaller prime of the form {z}1)
* zy{z} (not minimal prime (start with b+1) if there is smaller prime of the form y{z})
* {z}yz (not minimal prime (start with b+1) if there is smaller prime of the form {z}y)
* {z0}z1 (almost cannot be minimal prime (start with b+1), since this is not simple family)

(in fact, there are no bases 2<=b<=1024 such that 7{0}1 is unsolved family, base 1004 is the last to drop at length 54849, also there are no bases 2<=b<=1024 such that {z}x is unsolved family, base 542 is the last to drop at length 1944)

sweety439 2021-02-15 16:46

Currently this project only has bases 2<=b<=16, I have plan to extend bases to 36 when all bases 2<=b<=16 have searched to length >=100K and all unsolved families are also found, and after extending bases to 36 and finding all minimal primes (start with b+1) with length <=100K and all unsolved families for all bases 2<=b<=36, I will extend bases to 64, then to 256 and 1024

The final goal of this project is solving all bases 2<=b<=1024 (i.e. finding all minimal primes (start with b+1) in all bases 2<=b<=1024 and proving that they are all such primes and proving the primality for all of them). Many of these primes have already been found but much more work is needed to find additional primes (the smallest primes in the unsolved families).

Solving all bases 2<=b<=1024 (i.e. finding all minimal primes (start with b+1) in all bases 2<=b<=1024 and proving that they are all such primes and proving the primality for all of them) is not possible but we aim to find many minimal primes (start with b+1) in bases 2<=b<=1024 (including all such primes with length <= 25K) and find all unsolved families in all bases 2<=b<=1024 and prove that all such primes not in current list (these primes should have length > 25K) for bases 2<=b<=1024 are in one of these unsolved families for the corresponding base b and proving the primality for many of the minimal primes (start with b+1) in bases 2<=b<=1024 (special forms (where * represents string of digits with length <= (1/3)*(length of the number)): *{0}1 can be proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality test[/URL], *{z} can be proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality test[/URL], for other forms, we can only use [URL="http://www.ellipsa.eu/index.html"]Primo[/URL] with [URL="https://primes.utm.edu/prove/prove4_2.html"]ECPP primality test[/URL] to prove the primality, and if the number is very large (say > 2^65536), the known primality tests for such a number are too inefficient to run, thus we can only resort to a [URL="https://primes.utm.edu/glossary/page.php?sort=PRP"]probable primality[/URL] test such as [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL] and [URL="https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html"]Baillie–PSW primality test[/URL], unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process to find divisors rather than using [URL="https://primes.utm.edu/glossary/xpage/TrialDivision.html"]trial division[/URL].

The unsolved families are of the form (a*b^n+c)/gcd(a+c,b-1) with a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1. Except the special case c = +-1 and gcd(a+c,b-1) = 1, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin test or a Baillie–PSW test, unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process to find divisors rather than using trial division. To do this, we made use of Geoffrey Reynolds’ [URL="https://www.bc-team.org/app.php/dlext/?cat=3"]srsieve[/URL] software. This program uses the baby-step giant-step algorithm to find all primes p which divide a*b^n+c where p and n lie in a specified range. Since this program cannot handle the general case (a*b^n+c)/gcd(a+c,b-1) when gcd(a+c,b-1) > 1 we only used it to sieve the sequence a*b^n+c for primes p not dividing gcd(a+c,b-1), and initialized the list of candidates to not include n for which there is some prime p dividing gcd(a+c,b-1) for which p divides (a*b^n+c)/gcd(a+c,b-1). The program had to be modified slightly to remove a check which would prevent it from running in the case when a, b, and c were all odd (since then 2 divides a*b^n+c, but 2 may not divide (a*b^n+c)/gcd(a+c,b-1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software [URL="http://jpenne.free.fr/index2.html"]LLR[/URL] by Jean Penne. Although undocumented, it is possible to run this program on numbers of the form (a*b^n+c)/gcd(a+c,b-1) when gcd(a+c,b-1)>1, so this program required no modifications (also, LLR can prove the primality for numbers of the form a*b^n+-1 (i.e. the special case c=+-1 and gcd(a+c,b-1)=1) with b^n>a, the case c=1 and gcd(a+c,b-1)=1 is corresponding to families *{0}1, and the case c=-1 and gcd(a+c,b-1)=1 is corresponding to families *{z}). A script was also written which allowed one to run srsieve while LLR was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by LLR it would be removed from the list of candidates. In the cases where the elements of M(Lb) could be proven prime rigorously, we employed [URL="http://www.ellipsa.eu/public/primo/primo.html"]Primo[/URL] by Marcel Martin, an elliptic curve primality proving implementation.

(this is exactly why base 25 family EF{O} is searched to higher length than other unsolved families in base 25 for the original minimal prime problem (i.e. prime > base is not required), since large primes in this family can be proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality test[/URL], their N+1 is 366*25^n and can be trivially 100% factored, while for other unsolved families (not only for base 25, but also for other bases) listed in [URL="https://cbright.myweb.cs.uwindsor.ca/talks/minimal-slides.pdf"]https://cbright.myweb.cs.uwindsor.ca/talks/minimal-slides.pdf[/URL], neither [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1[/URL] nor [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1[/URL] can be trivially 100% factored, see [URL="https://github.com/curtisbright/mepn-data/commits/master?after=dfd73217eb03e6889e63769eda77bcf739922ef3+104&branch=master"]https://github.com/curtisbright/mepn-data/commits/master?after=dfd73217eb03e6889e63769eda77bcf739922ef3+104&branch=master[/URL] and [URL="https://github.com/curtisbright/mepn-data/commit/19af47f73bfe06a2dcad14fcc5b6fef8327cc01c"]https://github.com/curtisbright/mepn-data/commit/19af47f73bfe06a2dcad14fcc5b6fef8327cc01c[/URL] and [URL="https://github.com/curtisbright/mepn-data/tree/EFO"]https://github.com/curtisbright/mepn-data/tree/EFO[/URL])

Our algorithm then proceeds as follows:

1. Let

M := {minimal primes in base b of length ≤ 3}

L :=

where x ≠ 0 and Y is the set of digits y such that xyz has no subword in M.

2. While L contains non-simple families:

(a) Explore each family of L, and update L.

(b) Examine each family of L:

i. Let w be the shortest string in the family. If w has a subword in M, then remove the family from L. If w represents a prime, then add w to M and remove the family from L.

ii. If possible, simplify the family.

iii. Check if the family can be proven to contain no primes > base, and if so then remove the family from L.

(c) As much as possible and update L; after each split examine the new families as in (b).

Links of the programs to solve the problem for this project:

Sieving programs for the simple families (families of the form *{?}*, where * represents any strings of digits (may be empty string), ? represents any digit) (the numbers in these families are of the form (a*b^n+c)/gcd(a+c,b-1) for fixed integers a>=1, b>=2 (b is exactly the base), c != 0, gcd(a,c) = 1, gcd(b,c) = 1) (we use srsieve to sieve the sequence a*b^n+c with primes not dividing gcd(a+c,b-1), and delete the n such that (a*b^n+c)/gcd(a+c,b-1) is not coprime to gcd(a+c,b-1)): [URL="https://web.archive.org/web/20101125185927/http://sites.google.com/site/geoffreywalterreynolds/programs/srsieve"]srsieve[/URL] (new link: [URL="http://www.rieselprime.de/dl/CRUS_pack.zip"]srsieve, sr1sieve, sr2sieve, PFGW, LLR[/URL] and [URL="https://mersenneforum.org/attachment.php?attachmentid=24520&d=1616003491"]srbsieve[/URL], also the [URL="https://www.bc-team.org/app.php/dlext/?cat=3"]BOINC Confederation[/URL] for srsieve, sr1sieve, sr2sieve, srbsieve) [URL="https://sourceforge.net/projects/mtsieve"]mtsieve[/URL]

Primality testing programs: [URL="https://sourceforge.net/projects/openpfgw/"]PFGW[/URL] [URL="http://jpenne.free.fr/index2.html"]LLR[/URL] [URL="http://www.ellipsa.eu/public/primo/primo.html"]primo[/URL] [URL="https://primes.utm.edu/programs/NewPGen/"]NewPGen[/URL] [URL="https://primes.utm.edu/programs/gallot/"]Proth[/URL] [URL="https://web.archive.org/web/20110128054640/http://pages.prodigy.net/chris_nash/primeform.html"]PrimeForm[/URL] (except in the special case c = +-1 and gcd(a+c,b-1) = 1, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin test or a Baillie–PSW test, unless a divisor of the number can be found (trial division))

Download these programs: [URL="http://www.rieselprime.de/dl/CRUS_pack.zip"]srsieve, sr1sieve, sr2sieve, PFGW, LLR[/URL] [URL="https://sourceforge.net/projects/mtsieve/files/latest/download"]mtsieve[/URL] [URL="https://sourceforge.net/projects/openpfgw/files/latest/download"]PFGW[/URL] [URL="http://jpenne.free.fr/llr3/llrcuda384linux64.zip"]LLR[/URL] ([URL="http://jpenne.free.fr/llr3/llrcuda384src.zip"]completed source for LLR[/URL]) [URL="http://www.ellipsa.eu/public/primo/files/primo-433-lx64.7z"]primo[/URL] [URL="https://primes.utm.edu/programs/NewPGen/newpgen.zip"]NewPGen[/URL] [URL="https://primes.utm.edu/programs/gallot/proth.exe"]Proth[/URL] [URL="https://web.archive.org/web/20110128054640/http://pages.prodigy.net/chris_nash/pform.zip"]PrimeForm[/URL]

Currently, only bases 2, 3, 4, 5, 6, 7, 8, 10, 12 are completely solved, the complete list of the minimal primes (start with b+1) in these bases are

[CODE]
base 2:

11

base 3:

12
21
111

base 4:

11
13
23
31
221

base 5:

12
21
23
32
34
43
104
111
131
133
313
401
414
3101
10103
14444
30301
33001
33331
44441
300031
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013

base 6:

11
15
21
25
31
35
45
51
4401
4441
40041

base 7:

14
16
23
25
32
41
43
52
56
61
65
113
115
131
133
155
212
221
304
313
335
344
346
364
445
515
533
535
544
551
553
1022
1051
1112
1202
1211
1222
2111
3031
3055
3334
3503
3505
3545
4504
4555
5011
5455
5545
5554
6034
6634
11111
11201
30011
30101
31001
31111
33001
33311
35555
40054
100121
150001
300053
351101
531101
1100021
33333301
5100000001
33333333333333331

base 8:

13
15
21
23
27
35
37
45
51
53
57
65
73
75
107
111
117
141
147
161
177
225
255
301
343
361
401
407
417
431
433
463
467
471
631
643
661
667
701
711
717
747
767
3331
3411
4043
4443
4611
5205
6007
6101
6441
6477
6707
6777
7461
7641
47777
60171
60411
60741
444641
500025
505525
3344441
4444477
5500525
5550525
55555025
444444441
744444441
77774444441
7777777777771
555555555555525
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447

base 10:

11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
227
251
257
277
281
349
409
449
499
521
557
577
587
727
757
787
821
827
857
877
881
887
991
2087
2221
5051
5081
5501
5581
5801
5851
6469
6949
8501
9001
9049
9221
9551
9649
9851
9949
20021
20201
50207
60649
80051
666649
946669
5200007
22000001
60000049
66000049
66600049
80555551
555555555551
5000000000000000000000000000027

base 12:

11
15
17
1B
25
27
31
35
37
3B
45
4B
51
57
5B
61
67
6B
75
81
85
87
8B
91
95
A7
AB
B5
B7
221
241
2A1
2B1
2BB
401
421
447
471
497
565
655
665
701
70B
721
747
771
77B
797
7A1
7BB
907
90B
9BB
A41
B21
B2B
2001
200B
202B
222B
229B
292B
299B
4441
4707
4777
6A05
6AA5
729B
7441
7B41
929B
9777
992B
9947
997B
9997
A0A1
A201
A605
A6A5
AA65
B001
B0B1
BB01
BB41
600A5
7999B
9999B
AAAA1
B04A1
B0B9B
BAA01
BAAA1
BB09B
BBBB1
44AAA1
A00065
BBBAA1
AAA0001
B00099B
AA000001
BBBBBB99B
B0000000000000000000000000009B
400000000000000000000000000000000000000077
[/CODE]

and the condensed table for these bases is:

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×bn+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4*8^221+17)/7
10 77 5(0^28)27 31 5*10^30+27
12 106 4(0^39)77 42 4*12^41+91
[/CODE]

Bases 2≤b≤1024 such that these families can be ruled out as contain no primes > b (using covering congruence, algebra factorization, or combine of them): (only list families which [B]must[/B] be minimal primes (start with b+1))

[CODE]
1{0}1
b == 1 mod 2: Finite covering set {2}
b = m^r with odd r>1: Sum-of-rth-powers factorization
1{0}2
b == 0 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
1{0}3
b == 1 mod 2: Finite covering set {2}
b == 0 mod 3: Finite covering set {3}
1{0}4
b == 0 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
b = m^4: Aurifeuillian factorization of x^4+4y^4
1{0}z
(none)
{1}
b = m^r with r>1: Difference-of-rth-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))
1{2}
b == 0 mod 2: Finite covering set {2}
b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)
1{3}
b == 0 mod 3: Finite covering set {3}
b such that b and 3(b+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
b == 1 mod 2 such that 3(b+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)
1{4}
b == 0 mod 2: Finite covering set {2}
b such that b and 4(b+3) are both squares: Difference-of-squares factorization
1{z}
(none)
2{0}1
b == 1 mod 3: Finite covering set {3}
2{0}3
b == 0 mod 3: Finite covering set {3}
b == 1 mod 5: Finite covering set {5}
{2}1
b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)
2{z}
b == 1 mod 2: Finite covering set {2}
3{0}1
b == 1 mod 2: Finite covering set {2}
3{0}2
b == 0 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
3{0}4
b == 0 mod 2: Finite covering set {2}
b == 1 mod 7: Finite covering set {7}
{3}1
b such that b and 3(2b+1) are both squares: Difference-of-squares factorization (such bases are 121)
3{z}
b == 1 mod 3: Finite covering set {3}
b == 14 mod 15: Finite covering set {3, 5}
b = m^2: Difference-of-squares factorization
b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)
4{0}1
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
b = m^4: Aurifeuillian factorization of x^4+4y^4
4{0}3
b == 0 mod 3: Finite covering set {3}
b == 1 mod 7: Finite covering set {7}
{4}1
b such that b and 4(3b+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)
4{z}
b == 1 mod 2: Finite covering set {2}
5{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
5{z}
b == 1 mod 5: Finite covering set {5}
b == 34 mod 35: Finite covering set {5, 7}
b = 6m^2 with m == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)
6{0}1
b == 1 mod 7: Finite covering set {7}
b == 34 mod 35: Finite covering set {5, 7}
6{z}
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
7{0}1
b == 1 mod 2: Finite covering set {2}
7{z}
b == 1 mod 7: Finite covering set {7}
b == 20 mod 21: Finite covering set {3, 7}
b == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
b = m^3: Difference-of-cubes factorization
8{0}1
b == 1 mod 3: Finite covering set {3}
b == 20 mod 21: Finite covering set {3, 7}
b == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
b = 467: Finite covering set {3, 5, 7, 19, 37}
b = 722: Finite covering set {3, 5, 13, 73, 109}
b = m^3: Sum-of-cubes factorization
b = 128: Cannot have primes since 7n+3 cannot be power of 2
8{z}
b == 1 mod 2: Finite covering set {2}
b = m^2: Difference-of-squares factorization
b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)
9{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
9{z}
b == 1 mod 3: Finite covering set {3}
b == 32 mod 33: Finite covering set {3, 11}
A{0}1
b == 1 mod 11: Finite covering set {11}
b == 32 mod 33: Finite covering set {3, 11}
A{z}
b == 1 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
B{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
b == 14 mod 15: Finite covering set {3, 5}
B{z}
b == 1 mod 11: Finite covering set {11}
b == 142 mod 143: Finite covering set {11, 13}
b = 307: Finite covering set {5, 11, 29}
b = 901: Finite covering set {7, 11, 13, 19}
C{0}1
b == 1 mod 13: Finite covering set {13}
b == 142 mod 143: Finite covering set {11, 13}
b = 296, 901: Finite covering set {7, 11, 13, 19}
b = 562, 828, 900: Finite covering set {7, 13, 19}
b = 563: Finite covering set {5, 7, 13, 19, 29}
b = 597: Finite covering set {5, 13, 29}
y{z}
(none)
{y}z
(none)
z{0}1
(none)
{z}1
(none)
{z}y
b == 0 mod 2: Finite covering set {2}
[/CODE]

Large known (probable) primes (length ≥10000) in these families: (Format: base (length)) (only list families which [B]must[/B] be minimal primes (start with b+1))

[CODE]
1{0}1
(none)

1{0}2
(none)

1{0}3
(none)

1{0}4
53 (13403)

113 (10647)

1{0}z
113 (20089)

123 (64371)

{1}
152 (270217)

184 (16703)

200 (17807)

311 (36497)

326 (26713)

331 (25033)

371 (15527)

485 (99523)

629 (32233)

649 (43987)

670 (18617)

684 (22573)

691 (62903)

693 (41189)

731 (15427)

752 (32833)

872 (10093)

932 (20431)

1{z}
107 (21911)

170 (166429)

278 (43909)

303 (40175)

383 (20957)

515 (58467)

522 (62289)

578 (129469)

590 (15527)

647 (21577)

662 (16591)

698 (127559)

704 (62035)

845 (39407)

938 (40423)

969 (24097)

989 (26869)

2{0}1
101 (192276)

206 (46206)

218 (333926)

236 (161230)

257 (12184)

305 (16808)

467 (126776)

578 (44166)

626 (174204)

695 (94626)

752 (26164)

788 (72918)

869 (49150)

887 (27772)

899 (15732)

932 (13644)

2{z}
432 (16003)

3{0}1
(none)

3{z}
72 (1119850)

212 (34414)

218 (23050)

270 (89662)

303 (198358)

312 (51566)

422 (21738)

480 (93610)

513 (38032)

527 (46074)

566 (23874)

650 (498102)

686 (16584)

758 (15574)

783 (12508)

800 (33838)

921 (98668)

947 (10056)

4{0}1
107 (32587)

227 (13347)

257 (160423)

355 (10990)

410 (144079)

440 (56087)

452 (14155)

482 (30691)

542 (15983)

579 (67776)

608 (20707)

635 (11723)

650 (96223)

679 (69450)

737 (269303)

740 (58043)

789 (149140)

797 (468703)

920 (103687)

934 (101404)

962 (84235)

4{z}
14 (19699)

68 (13575)

254 (15451)

800 (20509)

5{0}1
326 (400786)

350 (20392)

554 (10630)

662 (13390)

926 (40036)

5{z}
258 (212135)

272 (148427)

299 (64898)

307 (26263)

354 (25566)

433 (283919)

635 (36163)

678 (40859)

692 (45447)

719 (20552)

768 (70214)

857 (23083)

867 (61411)

972 (36703)

6{0}1
108 (16318)

129 (16797)

409 (369833)

522 (52604)

587 (24120)

643 (164916)

762 (11152)

789 (27297)

986 (21634)

6{z}
68 (25396)

332 (15222)

338 (42868)

362 (146342)

488 (33164)

566 (164828)

980 (50878)

986 (12506)

1016 (23336)

7{0}1
398 (17473)

1004 (54849)

7{z}
97 (192336)

170 (15423)

194 (38361)

202 (155772)

282 (21413)

283 (164769)

332 (13205)

412 (29792)

560 (19905)

639 (10668)

655 (53009)

811 (31784)

814 (17366)

866 (108591)

908 (61797)

962 (31841)

992 (10605)

997 (15815)

8{0}1
23 (119216)

53 (227184)

158 (123476)

254 (67716)

320 (52004)

410 (279992)

425 (94662)

513 (19076)

518 (11768)

596 (148446)

641 (87702)

684 (23387)

695 (39626)

788 (11408)

893 (86772)

908 (243440)

920 (107822)

962 (47222)

998 (81240)

1013 (43872)

8{z}
138 (35686)

412 (12154)

788 (11326)

990 (23032)

9{0}1
248 (39511)

592 (96870)

9{z}
431 (43574)

446 (152028)

458 (126262)

599 (11776)

846 (12781)

A{0}1
173 (264235)

198 (47665)

311 (314807)

341 (106009)

449 (18507)

492 (42843)

605 (12395)

708 (17563)

710 (31039)

743 (285479)

786 (68169)

800 (15105)

802 (149320)

879 (25004)

929 (13065)

977 (125873)

986 (48279)

1004 (10645)

A{z}
368 (10867)

488 (10231)

534 (80328)

662 (13307)

978 (14066)

B{0}1
710 (15272)

740 (33520)

878 (227482)

B{z}
153 (21660)

186 (112718)

439 (18752)

593 (16064)

602 (36518)

707 (10573)

717 (67707)

C{0}1
68 (656922)

219 (29231)

230 (94751)

312 (21163)

334 (83334)

353 (20262)

359 (61295)

457 (10024)

481 (45941)

501 (20140)

593 (42779)

600 (11242)

604 (17371)

641 (26422)

700 (91953)

887 (13961)

919 (45359)

923 (64365)

992 (10300)

y{z}
38 (136212)

83 (21496)

113 (286644)

188 (13508)

401 (103670)

417 (21003)

458 (46900)

494 (21580)

518 (129372)

527 (65822)

602 (17644)

608 (36228)

638 (74528)

663 (47557)

723 (24536)

758 (50564)

833 (12220)

904 (13430)

938 (50008)

950 (16248)

z{0}1
202 (46774)

251 (102979)

272 (16681)

297 (14314)

298 (60671)

326 (64757)

347 (69661)

363 (142877)

452 (71941)

543 (10042)

564 (38065)

634 (84823)

788 (13541)

869 (12289)

890 (37377)

953 (60995)

1004 (29685)

{z}1
(none)

{z}y
317 (13896)
[/CODE]

Bases 2≤b≤1024 which have these families as unsolved families (unsolved families are families which are neither primes (>b) found nor can be proven to contain no primes > b): (only list families which [B]must[/B] be minimal primes (start with b+1))

[CODE]
1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥223)

1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)

1{0}3: 646, 718, 998 (length limit: 2000)

1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)

1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)

{1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000)

1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)

1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 (length limit: 2000)

1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)

1{z}: 581, 992, 1019 (length limit: ≥100000)

2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)

2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)

{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)

2{z}: 588, 972 (length limit: ≥100000)

3{0}1: 718, 912 (length limit: ≥100000)

3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)

3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)

{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)

3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)

4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)

4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)

{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)

4{z}: 338, 998 (length limit: ≥100000)

5{0}1: 308, 512, 824 (length limit: ≥100000)

5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)

6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)

6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)

7{0}1: (none)

7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)

8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 785, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)

8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)

9{0}1: 724, 884 (length limit: ≥100000)

9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)

A{0}1: 185, 338, 417, 432, 614, 668, 744, 773, 863, 935, 1000 (length limit: ≥100000)

A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)

B{0}1: 560, 770, 968 (length limit: ≥100000)

B{z}: 263, 615, 912, 978 (length limit: ≥100000)

C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)

{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)

y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)

z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)

{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)

{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)

{z}x: (none)

{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)
[/CODE]

[URL="http://www.noprimeleftbehind.net/crus/"]CRUS[/URL] found many minimal primes (start with b+1) in bases 2<=b<=1024, these primes are in families either *{0}1 or *{z} (where * represents any string of digits) for the corresponding base b

List of the length of the minimal primes (start with b+1) in given family for bases 2<=b<=1024 (only list families which [B]must[/B] be minimal primes (start with b+1)): [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL] ("RC" means this family can be ruled out as only contain composite numbers (only count numbers > base), "NB" means this family is not interpretable in this base (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base), "unknown" means this family is unsolved family)

More information of minimal primes (start with b+1) in given family for bases 2<=b<=1024 (only list families which [B]must[/B] be minimal primes (start with b+1)): [URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1017467222"]https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1017467222[/URL]

sweety439 2021-02-16 08:07

1 Attachment(s)
References of given simple families for the minimal primes (start with b+1) problem in bases 2<=b<=1024:

{1}:

[URL="http://www.users.globalnet.co.uk/~aads/primes.html"]http://www.users.globalnet.co.uk/~aads/primes.html[/URL] (broken link: [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]from wayback machine[/URL])
[URL="http://www.users.globalnet.co.uk/~aads/titans.html"]http://www.users.globalnet.co.uk/~aads/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html"]from wayback machine[/URL])
[URL="http://www.primes.viner-steward.org/andy/titans.html"]http://www.primes.viner-steward.org/andy/titans.html[/URL] (broken link: [URL="https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html"]from wayback machine[/URL])
[URL="http://www.phi.redgolpe.com/"]http://www.phi.redgolpe.com/[/URL] (broken link: [URL="https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/"]from wayback machine[/URL])
[URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]
[URL="https://oeis.org/A128164/a128164_7.txt"]https://oeis.org/A128164/a128164_7.txt[/URL]
[URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL]
[URL="http://www.mersennewiki.org/index.php/Repunit"]http://www.mersennewiki.org/index.php/Repunit[/URL] (broken link: [URL="https://web.archive.org/web/20180416000002/http://www.mersennewiki.org/index.php/Repunit"]from wayback machine[/URL])
[URL="https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf"]https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf[/URL]
[URL="http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470"]http://bbs.mathchina.com/bbs/forum.php?mod=viewthread&tid=2050470[/URL]
[URL="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906"]https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906[/URL] (archive today cannot automatically return the archive page, if you use archive today, click [URL="https://archive.is/WCvbi"]https://archive.is/WCvbi[/URL])
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://ebisui-hirotaka.com/img/file410.pdf"]http://ebisui-hirotaka.com/img/file410.pdf[/URL]
[URL="https://www.jstor.org/stable/2006470?origin=crossref"]https://www.jstor.org/stable/2006470?origin=crossref[/URL]
[URL="http://www.bitman.name/math/table/379"]http://www.bitman.name/math/table/379[/URL]
[URL="https://oeis.org/A084740"]https://oeis.org/A084740[/URL]
[URL="https://oeis.org/A084738"]https://oeis.org/A084738[/URL] (corresponding primes)
[URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL] (prime bases)
[URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL] (prime bases, corresponding primes)
[URL="https://oeis.org/A128164"]https://oeis.org/A128164[/URL] (length 2 not allowed)
[URL="https://oeis.org/A285642"]https://oeis.org/A285642[/URL] (length 2 not allowed, corresponding primes)

1{0}1:

[URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL]
[URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/index.html"]http://yves.gallot.pagesperso-orange.fr/primes/index.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/results.html"]http://yves.gallot.pagesperso-orange.fr/primes/results.html[/URL]
[URL="http://yves.gallot.pagesperso-orange.fr/primes/stat.html"]http://yves.gallot.pagesperso-orange.fr/primes/stat.html[/URL]
[URL="https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf"]https://www.ams.org/journals/mcom/2002-71-238/S0025-5718-01-01350-3/S0025-5718-01-01350-3.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf"]https://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0124264-0/S0025-5718-1961-0124264-0.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf"]https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917833-8/S0025-5718-1988-0917833-8.pdf[/URL] (b=2^n)
[URL="https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf"]https://www.ams.org/journals/mcom/1995-64-210/S0025-5718-1995-1277765-9/S0025-5718-1995-1277765-9.pdf[/URL] (b=2^n)
[URL="https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf"]https://www.sciencedirect.com/science/article/pii/S0022314X02927824/pdf?md5=7e215fd8dadaf84646ab82f2a96ebb8c&pid=1-s2.0-S0022314X02927824-main.pdf[/URL] (b=2^n)
[URL="https://arxiv.org/pdf/1605.01371.pdf"]https://arxiv.org/pdf/1605.01371.pdf[/URL] (b=2^n)
[URL="https://oeis.org/A228101"]https://oeis.org/A228101[/URL]
[URL="https://oeis.org/A079706"]https://oeis.org/A079706[/URL]
[URL="https://oeis.org/A084712"]https://oeis.org/A084712[/URL] (corresponding primes)
[URL="https://oeis.org/A123669"]https://oeis.org/A123669[/URL] (length 2 not allowed, corresponding primes)

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1:

[URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL] (2{0}1 in base 512, 4{0}1 in bases 32, 512, 1024, which are not in the first 4 references)
[URL="http://www.prothsearch.com/GFN10.html"]http://www.prothsearch.com/GFN10.html[/URL] (A{0}1 in base 1000, which are not in the first 4 references)
[URL="https://mersenneforum.org/showthread.php?t=6918"]https://mersenneforum.org/showthread.php?t=6918[/URL] (2{0}1)
[URL="https://mersenneforum.org/showthread.php?t=19725"]https://mersenneforum.org/showthread.php?t=19725[/URL] (2{0}1 in bases == 11 mod 12)
[URL="https://oeis.org/A119624"]https://oeis.org/A119624[/URL] (2{0}1)
[URL="https://oeis.org/A253178"]https://oeis.org/A253178[/URL] (2{0}1)
[URL="https://oeis.org/A098872"]https://oeis.org/A098872[/URL] (2{0}1 in bases divisible by 6)
[URL="https://oeis.org/A098877"]https://oeis.org/A098877[/URL] (3{0}1 in bases divisible by 6)
[URL="https://oeis.org/A088782"]https://oeis.org/A088782[/URL] (A{0}1)
[URL="https://oeis.org/A088622"]https://oeis.org/A088622[/URL] (A{0}1, corresponding primes)

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}:

[URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10354"]https://mersenneforum.org/showthread.php?t=10354[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719"]https://mersenneforum.org/attachment.php?attachmentid=23215&d=1598786719[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23844&d=1606276304[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304"]https://mersenneforum.org/attachment.php?attachmentid=23845&d=1606276304[/URL]
[URL="https://mersenneforum.org/showthread.php?t=24576"]https://mersenneforum.org/showthread.php?t=24576[/URL] (1{z})
[URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL] (1{z})
[URL="https://oeis.org/A119591"]https://oeis.org/A119591[/URL] (1{z})
[URL="https://oeis.org/A098873"]https://oeis.org/A098873[/URL] (1{z} in bases divisible by 6)
[URL="https://oeis.org/A098876"]https://oeis.org/A098876[/URL] (2{z} in bases divisible by 6)

z{0}1:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MP_table"]https://www.rieselprime.de/ziki/Williams_prime_MP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.prothsearch.com/riesel1a.html"]http://www.prothsearch.com/riesel1a.html[/URL] (base 512)
[URL="http://www.bitman.name/math/table/477"]http://www.bitman.name/math/table/477[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A305531"]https://oeis.org/A305531[/URL]
[URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL] (prime bases)

y{z}:

[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_MM_table"]https://www.rieselprime.de/ziki/Williams_prime_MM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL]
[URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL]
[URL="http://www.prothsearch.com/riesel2.html"]http://www.prothsearch.com/riesel2.html[/URL] (base 128)
[URL="http://www.bitman.name/math/table/484"]http://www.bitman.name/math/table/484[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL] (prime bases)

1{0}2:

[URL="https://oeis.org/A138066"]https://oeis.org/A138066[/URL]
[URL="https://oeis.org/A084713"]https://oeis.org/A084713[/URL] (corresponding primes)
[URL="https://oeis.org/A138067"]https://oeis.org/A138067[/URL] (length 2 not allowed)

1{0}z:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]
[URL="https://oeis.org/A076846"]https://oeis.org/A076846[/URL] (corresponding primes)
[URL="https://oeis.org/A078178"]https://oeis.org/A078178[/URL] (length 2 not allowed)
[URL="https://oeis.org/A078179"]https://oeis.org/A078179[/URL] (length 2 not allowed, corresponding primes)

{z}1:

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/435"]http://www.bitman.name/math/table/435[/URL] (prime bases)
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL]
[URL="https://oeis.org/A343589"]https://oeis.org/A343589[/URL] (corresponding primes)
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"]https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html[/URL] (prime bases)

11{0}1: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_least"]https://www.rieselprime.de/ziki/Williams_prime_PP_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PP_table"]https://www.rieselprime.de/ziki/Williams_prime_PP_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/474"]http://www.bitman.name/math/table/474[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

1{0}11: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A346149"]https://oeis.org/A346149[/URL]
[URL="https://oeis.org/A346154"]https://oeis.org/A346154[/URL] (corresponding primes)

10{z}: (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})

[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_least"]https://www.rieselprime.de/ziki/Williams_prime_PM_least[/URL]
[URL="https://www.rieselprime.de/ziki/Williams_prime_PM_table"]https://www.rieselprime.de/ziki/Williams_prime_PM_table[/URL]
[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="http://www.bitman.name/math/table/471"]http://www.bitman.name/math/table/471[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]

{z}y:

[URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] (length 1 allowed)
[URL="https://oeis.org/A250200"]https://oeis.org/A250200[/URL]
[URL="https://oeis.org/A255707"]https://oeis.org/A255707[/URL] (length 1 allowed)
[URL="https://oeis.org/A084714"]https://oeis.org/A084714[/URL] (length 1 allowed, corresponding primes)
[URL="https://oeis.org/A292201"]https://oeis.org/A292201[/URL] (length 1 allowed, prime bases)

{z}yz: (not minimal prime (start with b+1) if there is smaller prime of the form {z}y)

[URL="https://sites.google.com/view/williams-primes"]https://sites.google.com/view/williams-primes[/URL]
[URL="https://mersenneforum.org/showthread.php?t=21818"]https://mersenneforum.org/showthread.php?t=21818[/URL]
[URL="https://oeis.org/A178250"]https://oeis.org/A178250[/URL]

{#}$: (for odd base b, # = (b−1)/2, $ = (b+1)/2)

[URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL]
[URL="http://www.prothsearch.com/GFN05.html"]http://www.prothsearch.com/GFN05.html[/URL] (base 625)

{z0}z1: (almost cannot be minimal prime (start with b+1), since this is not simple family, but always minimal prime (start with b'+1) in base b'=b^2)

[URL="http://www.fermatquotient.com/PrimSerien/GenRepuP.txt"]http://www.fermatquotient.com/PrimSerien/GenRepuP.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171"]https://mersenneforum.org/attachment.php?attachmentid=19026&d=1535956171[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf[/URL]
[URL="http://www.primenumbers.net/Henri/us/MersFermus.htm"]http://www.primenumbers.net/Henri/us/MersFermus.htm[/URL]
[URL="http://www.bitman.name/math/table/488"]http://www.bitman.name/math/table/488[/URL]
[URL="https://oeis.org/A084742"]https://oeis.org/A084742[/URL]
[URL="https://oeis.org/A084741"]https://oeis.org/A084741[/URL] (corresponding primes)
[URL="https://oeis.org/A065507"]https://oeis.org/A065507[/URL] (prime bases)

OEIS sequences (only list those for families ?{?}, {?}?, ?{0}?, since they [B]must[/B] be minimal primes (start with b+1)):

Base 2:

[URL="https://oeis.org/A000043"]https://oeis.org/A000043[/URL] ({1})
[URL="https://oeis.org/A000668"]https://oeis.org/A000668[/URL] ({1}, corresponding primes)
[URL="https://oeis.org/A019434"]https://oeis.org/A019434[/URL] (1{0}1, corresponding primes)

Base 3:

[URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL] ({1})
[URL="https://oeis.org/A076481"]https://oeis.org/A076481[/URL] ({1}, corresponding primes)
[URL="https://oeis.org/A003307"]https://oeis.org/A003307[/URL] (1{2})
[URL="https://oeis.org/A079363"]https://oeis.org/A079363[/URL] (1{2}, corresponding primes)
[URL="https://oeis.org/A171381"]https://oeis.org/A171381[/URL] ({1}2)
[URL="https://oeis.org/A093625"]https://oeis.org/A093625[/URL] ({1}2, corresponding primes)
[URL="https://oeis.org/A014224"]https://oeis.org/A014224[/URL] ({2}1)
[URL="https://oeis.org/A014232"]https://oeis.org/A014232[/URL] ({2}1, corresponding primes)
[URL="https://oeis.org/A051783"]https://oeis.org/A051783[/URL] (1{0}2)
[URL="https://oeis.org/A057735"]https://oeis.org/A057735[/URL] (1{0}2, corresponding primes)
[URL="https://oeis.org/A003306"]https://oeis.org/A003306[/URL] (2{0}1)
[URL="https://oeis.org/A111974"]https://oeis.org/A111974[/URL] (2{0}1, corresponding primes)

Base 4:

[URL="https://oeis.org/A146768"]https://oeis.org/A146768[/URL] (1{3})
[URL="https://oeis.org/A000668"]https://oeis.org/A000668[/URL] (1{3}, corresponding primes)
[URL="https://oeis.org/A261539"]https://oeis.org/A261539[/URL] ({1}3)
[URL="https://oeis.org/A272057"]https://oeis.org/A272057[/URL] (2{3})
[URL="https://oeis.org/A127936"]https://oeis.org/A127936[/URL] ({2}3)
[URL="https://oeis.org/A000979"]https://oeis.org/A000979[/URL] ({2}3, corresponding primes)
[URL="https://oeis.org/A059266"]https://oeis.org/A059266[/URL] ({3}1)
[URL="https://oeis.org/A135535"]https://oeis.org/A135535[/URL] ({3}1, corresponding primes)
[URL="https://oeis.org/A222008"]https://oeis.org/A222008[/URL] (1{0}1, corresponding primes)
[URL="https://oeis.org/A089437"]https://oeis.org/A089437[/URL] (1{0}3)
[URL="https://oeis.org/A228026"]https://oeis.org/A228026[/URL] (1{0}3, corresponding primes)
[URL="https://oeis.org/A326655"]https://oeis.org/A326655[/URL] (3{0}1)

Base 5:

[URL="https://oeis.org/A004061"]https://oeis.org/A004061[/URL] ({1})
[URL="https://oeis.org/A086122"]https://oeis.org/A086122[/URL] ({1}, corresponding primes)
[URL="https://oeis.org/A120375"]https://oeis.org/A120375[/URL] (1{4})
[URL="https://oeis.org/A120376"]https://oeis.org/A120376[/URL] (1{4}, corresponding primes)
[URL="https://oeis.org/A046865"]https://oeis.org/A046865[/URL] (3{4})
[URL="https://oeis.org/A059613"]https://oeis.org/A059613[/URL] ({4}1)
[URL="https://oeis.org/A181285"]https://oeis.org/A181285[/URL] ({4}1, corresponding primes)
[URL="https://oeis.org/A109080"]https://oeis.org/A109080[/URL] ({4}3)
[URL="https://oeis.org/A204578"]https://oeis.org/A204578[/URL] ({4}3, corresponding primes)
[URL="https://oeis.org/A087885"]https://oeis.org/A087885[/URL] (1{0}2)
[URL="https://oeis.org/A182330"]https://oeis.org/A182330[/URL] (1{0}2, corresponding primes)
[URL="https://oeis.org/A124621"]https://oeis.org/A124621[/URL] (1{0}4)
[URL="https://oeis.org/A228028"]https://oeis.org/A228028[/URL] (1{0}4, corresponding primes)
[URL="https://oeis.org/A058934"]https://oeis.org/A058934[/URL] (2{0}1)
[URL="https://oeis.org/A205771"]https://oeis.org/A205771[/URL] (2{0}1, corresponding primes)
[URL="https://oeis.org/A204322"]https://oeis.org/A204322[/URL] (4{0}1)

Base 6:

[URL="https://oeis.org/A004062"]https://oeis.org/A004062[/URL] ({1})
[URL="https://oeis.org/A165210"]https://oeis.org/A165210[/URL] ({1}, corresponding primes)
[URL="https://oeis.org/A057472"]https://oeis.org/A057472[/URL] (1{5})
[URL="https://oeis.org/A319535"]https://oeis.org/A319535[/URL] (1{5}, corresponding primes)
[URL="https://oeis.org/A186106"]https://oeis.org/A186106[/URL] (2{5})
[URL="https://oeis.org/A186104"]https://oeis.org/A186104[/URL] (2{5}, corresponding primes)
[URL="https://oeis.org/A079906"]https://oeis.org/A079906[/URL] (4{5})
[URL="https://oeis.org/A248613"]https://oeis.org/A248613[/URL] ({4}5)
[URL="https://oeis.org/A059614"]https://oeis.org/A059614[/URL] ({5}1)
[URL="https://oeis.org/A290008"]https://oeis.org/A290008[/URL] ({5}1, corresponding primes)
[URL="https://oeis.org/A182331"]https://oeis.org/A182331[/URL] (1{0}1, corresponding primes)
[URL="https://oeis.org/A145106"]https://oeis.org/A145106[/URL] (1{0}5)
[URL="https://oeis.org/A104118"]https://oeis.org/A104118[/URL] (1{0}5, corresponding primes)
[URL="https://oeis.org/A120023"]https://oeis.org/A120023[/URL] (2{0}1)
[URL="https://oeis.org/A205776"]https://oeis.org/A205776[/URL] (2{0}1, corresponding primes)
[URL="https://oeis.org/A186112"]https://oeis.org/A186112[/URL] (3{0}1)
[URL="https://oeis.org/A186105"]https://oeis.org/A186105[/URL] (3{0}1, corresponding primes)
[URL="https://oeis.org/A247260"]https://oeis.org/A247260[/URL] (5{0}1)

Base 10:

[URL="https://oeis.org/A004023"]https://oeis.org/A004023[/URL] ({1})
[URL="https://oeis.org/A004022"]https://oeis.org/A004022[/URL] ({1}, corresponding primes)
[URL="https://oeis.org/A056698"]https://oeis.org/A056698[/URL] (1{3})
[URL="https://oeis.org/A093671"]https://oeis.org/A093671[/URL] (1{3}, corresponding primes)
[URL="https://oeis.org/A097683"]https://oeis.org/A097683[/URL] ({1}3)
[URL="https://oeis.org/A093011"]https://oeis.org/A093011[/URL] ({1}3, corresponding primes)
[URL="https://oeis.org/A089147"]https://oeis.org/A089147[/URL] (1{7})
[URL="https://oeis.org/A088465"]https://oeis.org/A088465[/URL] (1{7}, corresponding primes)
[URL="https://oeis.org/A097684"]https://oeis.org/A097684[/URL] ({1}7)
[URL="https://oeis.org/A093139"]https://oeis.org/A093139[/URL] ({1}7, corresponding primes)
[URL="https://oeis.org/A002957"]https://oeis.org/A002957[/URL] (1{9})
[URL="https://oeis.org/A055558"]https://oeis.org/A055558[/URL] (1{9}, corresponding primes)
[URL="https://oeis.org/A097685"]https://oeis.org/A097685[/URL] ({1}9)
[URL="https://oeis.org/A093400"]https://oeis.org/A093400[/URL] ({1}9, corresponding primes)
[URL="https://oeis.org/A056700"]https://oeis.org/A056700[/URL] (2{1})
[URL="https://oeis.org/A068814"]https://oeis.org/A068814[/URL] (2{1}, corresponding primes)
[URL="https://oeis.org/A084832"]https://oeis.org/A084832[/URL] ({2}1)
[URL="https://oeis.org/A091189"]https://oeis.org/A091189[/URL] ({2}1, corresponding primes)
[URL="https://oeis.org/A056701"]https://oeis.org/A056701[/URL] (2{3})
[URL="https://oeis.org/A093672"]https://oeis.org/A093672[/URL] (2{3}, corresponding primes)
[URL="https://oeis.org/A096506"]https://oeis.org/A096506[/URL] ({2}3)
[URL="https://oeis.org/A093162"]https://oeis.org/A093162[/URL] ({2}3, corresponding primes)
[URL="https://oeis.org/A056702"]https://oeis.org/A056702[/URL] (2{7})
[URL="https://oeis.org/A093938"]https://oeis.org/A093938[/URL] (2{7}, corresponding primes)
[URL="https://oeis.org/A099409"]https://oeis.org/A099409[/URL] ({2}7)
[URL="https://oeis.org/A093167"]https://oeis.org/A093167[/URL] ({2}7, corresponding primes)
[URL="https://oeis.org/A056703"]https://oeis.org/A056703[/URL] (2{9})
[URL="https://oeis.org/A055559"]https://oeis.org/A055559[/URL] (2{9}, corresponding primes)
[URL="https://oeis.org/A099410"]https://oeis.org/A099410[/URL] ({2}9)
[URL="https://oeis.org/A093401"]https://oeis.org/A093401[/URL] ({2}9, corresponding primes)
[URL="https://oeis.org/A056704"]https://oeis.org/A056704[/URL] (3{1})
[URL="https://oeis.org/A068813"]https://oeis.org/A068813[/URL] (3{1}, corresponding primes)
[URL="https://oeis.org/A055557"]https://oeis.org/A055557[/URL] ({3}1)
[URL="https://oeis.org/A123568"]https://oeis.org/A123568[/URL] ({3}1, corresponding primes)
[URL="https://oeis.org/A056705"]https://oeis.org/A056705[/URL] (3{7})
[URL="https://oeis.org/A093939"]https://oeis.org/A093939[/URL] (3{7}, corresponding primes)
[URL="https://oeis.org/A056680"]https://oeis.org/A056680[/URL] ({3}7)
[URL="https://oeis.org/A093168"]https://oeis.org/A093168[/URL] ({3}7, corresponding primes)
[URL="https://oeis.org/A049054"]https://oeis.org/A049054[/URL] (1{0}3)
[URL="https://oeis.org/A159352"]https://oeis.org/A159352[/URL] (1{0}3, corresponding primes)
[URL="https://oeis.org/A088274"]https://oeis.org/A088274[/URL] (1{0}7)
[URL="https://oeis.org/A159031"]https://oeis.org/A159031[/URL] (1{0}7, corresponding primes)
[URL="https://oeis.org/A088275"]https://oeis.org/A088275[/URL] (1{0}9)
[URL="https://oeis.org/A081677"]https://oeis.org/A081677[/URL] (2{0}3)
[URL="https://oeis.org/A177134"]https://oeis.org/A177134[/URL] (2{0}3, corresponding primes)
[URL="https://oeis.org/A101392"]https://oeis.org/A101392[/URL] (2{0}9)
[URL="https://oeis.org/A056807"]https://oeis.org/A056807[/URL] (3{0}1)
[URL="https://oeis.org/A259866"]https://oeis.org/A259866[/URL] (3{0}1, corresponding primes)
[URL="https://oeis.org/A100501"]https://oeis.org/A100501[/URL] (3{0}7)
[URL="https://oeis.org/A056806"]https://oeis.org/A056806[/URL] (4{0}1)
[URL="https://oeis.org/A177506"]https://oeis.org/A177506[/URL] (4{0}1, corresponding primes)
[URL="https://oeis.org/A101397"]https://oeis.org/A101397[/URL] (4{0}3)
[URL="https://oeis.org/A177507"]https://oeis.org/A177507[/URL] (4{0}3, corresponding primes)
[URL="https://oeis.org/A101395"]https://oeis.org/A101395[/URL] (4{0}7)
[URL="https://oeis.org/A101394"]https://oeis.org/A101394[/URL] (4{0}9)

OEIS sequences for [I]smallest base yielding primes[/I] for given length:

[URL="https://oeis.org/A066180"]https://oeis.org/A066180[/URL] ({1})
[URL="https://oeis.org/A056993"]https://oeis.org/A056993[/URL] (1{0}1)
[URL="https://oeis.org/A113517"]https://oeis.org/A113517[/URL] ({z}1)
[URL="https://oeis.org/A248079"]https://oeis.org/A248079[/URL] (1{0}z)
[URL="https://oeis.org/A157922"]https://oeis.org/A157922[/URL] (1{z})
[URL="https://oeis.org/A127599"]https://oeis.org/A127599[/URL] ({z}yz)
[URL="https://oeis.org/A275530"]https://oeis.org/A275530[/URL] ({#}$: (for odd base b, # = (b−1)/2, $ = (b+1)/2))
[URL="https://oeis.org/A103795"]https://oeis.org/A103795[/URL] ({z0}z1)

Other references:

[URL="https://oeis.org/"]https://oeis.org/[/URL]
[URL="http://factordb.com/"]http://factordb.com/[/URL]
[URL="https://primes.utm.edu/"]https://primes.utm.edu/[/URL]
[URL="https://primes.utm.edu/primes/"]https://primes.utm.edu/primes/[/URL]
[URL="https://primes.utm.edu/primes/download.php"]https://primes.utm.edu/primes/download.php[/URL] (archive today cannot automatically return the archive page, if you use archive today, click [URL="https://archive.ph/BvS68"]https://archive.ph/BvS68[/URL])
[URL="https://primes.utm.edu/primes/lists/all.txt"]https://primes.utm.edu/primes/lists/all.txt[/URL]
[URL="https://primes.utm.edu/largest.html"]https://primes.utm.edu/largest.html[/URL]
[URL="https://primes.utm.edu/top20/index.php"]https://primes.utm.edu/top20/index.php[/URL]
[URL="https://primes.utm.edu/prove/index.html"]https://primes.utm.edu/prove/index.html[/URL]
[URL="https://primes.utm.edu/mersenne/"]https://primes.utm.edu/mersenne/[/URL]
[URL="https://primes.utm.edu/notes/proofs/"]https://primes.utm.edu/notes/proofs/[/URL]
[URL="http://www.primenumbers.net/prptop/prptop.php"]http://www.primenumbers.net/prptop/prptop.php[/URL]
[URL="https://primes.utm.edu/primes/search.php"]https://primes.utm.edu/primes/search.php[/URL]
[URL="http://www.primenumbers.net/prptop/searchform.php"]http://www.primenumbers.net/prptop/searchform.php[/URL]
[URL="http://www.ellipsa.eu/public/primo/top20.html"]http://www.ellipsa.eu/public/primo/top20.html[/URL]
[URL="http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html"]http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html[/URL]
[URL="http://www.primegrid.com"]http://www.primegrid.com[/URL]
[URL="https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics)"]https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics)[/URL]
[URL="https://en.wikiversity.org/wiki/Quasi-minimal_prime"]https://en.wikiversity.org/wiki/Quasi-minimal_prime[/URL]
[URL="https://primes.utm.edu/glossary/xpage/MinimalPrime.html"]https://primes.utm.edu/glossary/xpage/MinimalPrime.html[/URL]
[URL="https://primes.utm.edu/curios/page.php?number_id=22380"]https://primes.utm.edu/curios/page.php?number_id=22380[/URL]
[URL="https://www.rieselprime.de/ziki/Main_Page"]https://www.rieselprime.de/ziki/Main_Page[/URL]
[URL="https://www.rose-hulman.edu/~rickert/Compositeseq/"]https://www.rose-hulman.edu/~rickert/Compositeseq/[/URL]
[URL="http://www.worldofnumbers.com/em197.htm"]http://www.worldofnumbers.com/em197.htm[/URL]
[URL="http://www.worldofnumbers.com/Appending%201s%20to%20n.txt"]http://www.worldofnumbers.com/Appending%201s%20to%20n.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=25000&d=1622618552"]https://mersenneforum.org/attachment.php?attachmentid=25000&d=1622618552[/URL]
[URL="http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm"]http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm[/URL] (broken link: [URL="https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm"]from wayback machine[/URL])
[URL="https://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.153"]https://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.153[/URL]
[URL="http://list.seqfan.eu/pipermail/seqfan/2014-September/013620.html"]http://list.seqfan.eu/pipermail/seqfan/2014-September/013620.html[/URL]
[URL="http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html"]http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html[/URL]
[URL="https://arxiv.org/pdf/1503.08883.pdf"]https://arxiv.org/pdf/1503.08883.pdf[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=15962&d=1492836533"]https://mersenneforum.org/attachment.php?attachmentid=15962&d=1492836533[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=15963&d=1492836533"]https://mersenneforum.org/attachment.php?attachmentid=15963&d=1492836533[/URL]
[URL="https://oeis.org/A069568"]https://oeis.org/A069568[/URL]
[URL="https://oeis.org/A112386"]https://oeis.org/A112386[/URL]
[URL="https://oeis.org/A083747"]https://oeis.org/A083747[/URL]
[URL="https://oeis.org/A090584"]https://oeis.org/A090584[/URL]
[URL="https://oeis.org/A090464"]https://oeis.org/A090464[/URL]
[URL="https://oeis.org/A090465"]https://oeis.org/A090465[/URL]
[URL="https://oeis.org/A257459"]https://oeis.org/A257459[/URL]
[URL="https://oeis.org/A232210"]https://oeis.org/A232210[/URL]
[URL="https://oeis.org/A257460"]https://oeis.org/A257460[/URL]
[URL="https://oeis.org/A257461"]https://oeis.org/A257461[/URL]
[URL="https://oeis.org/A200065"]https://oeis.org/A200065[/URL]
[URL="https://oeis.org/A086766"]https://oeis.org/A086766[/URL]
[URL="https://oeis.org/A087403"]https://oeis.org/A087403[/URL]
[URL="https://oeis.org/A272232"]https://oeis.org/A272232[/URL]
[URL="https://oeis.org/A306861"]https://oeis.org/A306861[/URL]
[URL="https://oeis.org/A307873"]https://oeis.org/A307873[/URL]
[URL="https://oeis.org/A256481"]https://oeis.org/A256481[/URL]
[URL="https://oeis.org/A090287"]https://oeis.org/A090287[/URL]
[URL="https://www.primepuzzles.net/puzzles/puzz_197.htm"]https://www.primepuzzles.net/puzzles/puzz_197.htm[/URL]
[URL="https://www.primepuzzles.net/puzzles/puzz_614.htm"]https://www.primepuzzles.net/puzzles/puzz_614.htm[/URL]
[URL="https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf"]https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf[/URL]
[URL="http://ostracodfiles.com/primes14/primes.php"]http://ostracodfiles.com/primes14/primes.php[/URL] (base 14, family *{D})
[URL="https://stdkmd.net/nrr/prime/"]https://stdkmd.net/nrr/prime/[/URL]
[URL="https://stdkmd.net/nrr/records.htm"]https://stdkmd.net/nrr/records.htm[/URL]
[URL="https://stdkmd.net/nrr/cert/"]https://stdkmd.net/nrr/cert/[/URL]
[URL="https://stdkmd.net/nrr/pock/"]https://stdkmd.net/nrr/pock/[/URL]
[URL="https://stdkmd.net/nrr/coveringset.htm"]https://stdkmd.net/nrr/coveringset.htm[/URL]
[URL="https://stdkmd.net/nrr/prime/primecount.txt"]https://stdkmd.net/nrr/prime/primecount.txt[/URL]
[URL="https://stdkmd.net/nrr/prime/primecount2.txt"]https://stdkmd.net/nrr/prime/primecount2.txt[/URL]
[URL="https://stdkmd.net/nrr/prime/primecount3.txt"]https://stdkmd.net/nrr/prime/primecount3.txt[/URL]
[URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]https://stdkmd.net/nrr/prime/primedifficulty.txt[/URL]
[URL="https://stdkmd.net/nrr/prime/primesize.txt"]https://stdkmd.net/nrr/prime/primesize.txt[/URL]
[URL="https://stdkmd.net/nrr/repunit/"]https://stdkmd.net/nrr/repunit/[/URL]
[URL="https://stdkmd.net/nrr/repunit/10001.htm"]https://stdkmd.net/nrr/repunit/10001.htm[/URL]
[URL="https://stdkmd.net/nrr/repunit/phin10.htm"]https://stdkmd.net/nrr/repunit/phin10.htm[/URL]
[URL="https://stdkmd.net/nrr/repunit/Phin10.txt"]https://stdkmd.net/nrr/repunit/Phin10.txt[/URL]
[URL="https://stdkmd.net/nrr/repunit/prpfactors.htm"]https://stdkmd.net/nrr/repunit/prpfactors.htm[/URL]
[URL="https://stdkmd.net/nrr/cert/Phi/"]https://stdkmd.net/nrr/cert/Phi/[/URL]
[URL="https://stdkmd.net/nrr/aaaab.htm"]https://stdkmd.net/nrr/aaaab.htm[/URL]
[URL="https://stdkmd.net/nrr/abbbb.htm"]https://stdkmd.net/nrr/abbbb.htm[/URL]
[URL="https://stdkmd.net/nrr/aaaba.htm"]https://stdkmd.net/nrr/aaaba.htm[/URL]
[URL="https://stdkmd.net/nrr/abaaa.htm"]https://stdkmd.net/nrr/abaaa.htm[/URL]
[URL="https://stdkmd.net/nrr/abbba.htm"]https://stdkmd.net/nrr/abbba.htm[/URL]
[URL="https://stdkmd.net/nrr/abbbc.htm"]https://stdkmd.net/nrr/abbbc.htm[/URL]
[URL="https://stdkmd.net/nrr/aabaa.htm"]https://stdkmd.net/nrr/aabaa.htm[/URL]
[URL="https://www.kurtbeschorner.de/"]https://www.kurtbeschorner.de/[/URL]
[URL="https://gmplib.org/~tege/repunit.html"]https://gmplib.org/~tege/repunit.html[/URL]
[URL="https://repunit-koide.jimdofree.com/"]https://repunit-koide.jimdofree.com/[/URL]
[URL="http://www.h4.dion.ne.jp/~rep/"]http://www.h4.dion.ne.jp/~rep/[/URL] (broken link: [URL="https://web.archive.org/web/20170120053509/http://www.h4.dion.ne.jp/~rep/"]from wayback machine[/URL])
[URL="http://repunit:1031@repunits.skoberne.net/list/"]http://repunit:1031@repunits.skoberne.net/list/[/URL] (broken link: [URL="https://web.archive.org/web/20170120055922/http://repunit:1031@repunits.skoberne.net/list/"]from wayback machine[/URL])
[URL="https://homes.cerias.purdue.edu/~ssw/cun/index.html"]https://homes.cerias.purdue.edu/~ssw/cun/index.html[/URL]
[URL="http://myfactors.mooo.com/"]http://myfactors.mooo.com/[/URL]
[URL="https://www.mersenne.org/"]https://www.mersenne.org/[/URL]
[URL="https://www.mersenne.ca/"]https://www.mersenne.ca/[/URL]
[URL="https://oeis.org/A250197/a250197_2.txt"]https://oeis.org/A250197/a250197_2.txt[/URL]
[URL="https://maths-people.anu.edu.au/~brent/factors.html"]https://maths-people.anu.edu.au/~brent/factors.html[/URL]
[URL="https://maths-people.anu.edu.au/~brent/pub/pub134.html"]https://maths-people.anu.edu.au/~brent/pub/pub134.html[/URL]
[URL="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm"]http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm[/URL]
[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119"]https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119[/URL]
[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339"]https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339[/URL]
[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039703851"]https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039703851[/URL]
[URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040202723"]https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040202723[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/bases%20b%20such%20that%20there%20is%20unique%20prime%20with%20period%20length%20n"]https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/bases%20b%20such%20that%20there%20is%20unique%20prime%20with%20period%20length%20n[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/unique%20period%20length%20in%20base%20b"]https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/unique%20period%20length%20in%20base%20b[/URL]
[URL="https://members.loria.fr/PZimmermann/ecmnet/"]https://members.loria.fr/PZimmermann/ecmnet/[/URL]
[URL="https://members.loria.fr/PZimmermann/records/ecmnet.html"]https://members.loria.fr/PZimmermann/records/ecmnet.html[/URL]
[URL="http://www.worldofnumbers.com/undulat.htm"]http://www.worldofnumbers.com/undulat.htm[/URL] (base 100)
[URL="http://www.worldofnumbers.com/deplat.htm"]http://www.worldofnumbers.com/deplat.htm[/URL]
[URL="http://www.worldofnumbers.com/wing.htm"]http://www.worldofnumbers.com/wing.htm[/URL]
[URL="http://www.worldofnumbers.com/merlon.htm"]http://www.worldofnumbers.com/merlon.htm[/URL]
[URL="http://www.primenumbers.net/Henri/us/NouvP1us.htm"]http://www.primenumbers.net/Henri/us/NouvP1us.htm[/URL]
[URL="https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31"]https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31[/URL] (base 38, family 1:{0}:31)
[URL="https://math.stackexchange.com/questions/760966/is-324455n-ever-prime"]https://math.stackexchange.com/questions/760966/is-324455n-ever-prime[/URL] (base 455, family 1:{0}:324)
[URL="http://www.prothsearch.com/sierp.html"]http://www.prothsearch.com/sierp.html[/URL]
[URL="http://www.prothsearch.com/rieselprob.html"]http://www.prothsearch.com/rieselprob.html[/URL]
[URL="https://www.rechenkraft.net/wiki/Seventeen_or_bust"]https://www.rechenkraft.net/wiki/Seventeen_or_bust[/URL]
[URL="https://oeis.org/A076336"]https://oeis.org/A076336[/URL]
[URL="https://oeis.org/A076337"]https://oeis.org/A076337[/URL]
[URL="https://oeis.org/A101036"]https://oeis.org/A101036[/URL]
[URL="https://oeis.org/A076335"]https://oeis.org/A076335[/URL]
[URL="https://oeis.org/A270271"]https://oeis.org/A270271[/URL]
[URL="https://oeis.org/A244561"]https://oeis.org/A244561[/URL]
[URL="https://oeis.org/A244070"]https://oeis.org/A244070[/URL]
[URL="https://oeis.org/A244562"]https://oeis.org/A244562[/URL]
[URL="https://oeis.org/A244071"]https://oeis.org/A244071[/URL]
[URL="https://oeis.org/A244563"]https://oeis.org/A244563[/URL]
[URL="https://oeis.org/A244072"]https://oeis.org/A244072[/URL]
[URL="https://oeis.org/A244564"]https://oeis.org/A244564[/URL]
[URL="https://oeis.org/A244073"]https://oeis.org/A244073[/URL]
[URL="https://oeis.org/A244565"]https://oeis.org/A244565[/URL]
[URL="https://oeis.org/A244074"]https://oeis.org/A244074[/URL]
[URL="https://oeis.org/A244566"]https://oeis.org/A244566[/URL]
[URL="https://oeis.org/A244076"]https://oeis.org/A244076[/URL]
[URL="https://oeis.org/A257647"]https://oeis.org/A257647[/URL]
[URL="https://oeis.org/A258074"]https://oeis.org/A258074[/URL]
[URL="https://oeis.org/A233469"]https://oeis.org/A233469[/URL]
[URL="https://oeis.org/A251057"]https://oeis.org/A251057[/URL]
[URL="https://oeis.org/A251757"]https://oeis.org/A251757[/URL]
[URL="https://oeis.org/A244549"]https://oeis.org/A244549[/URL]
[URL="https://oeis.org/A244351"]https://oeis.org/A244351[/URL]
[URL="https://oeis.org/A244545"]https://oeis.org/A244545[/URL]
[URL="https://oeis.org/A244211"]https://oeis.org/A244211[/URL]
[URL="https://oeis.org/A243969"]https://oeis.org/A243969[/URL]
[URL="https://oeis.org/A243974"]https://oeis.org/A243974[/URL]
[URL="https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf"]https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf[/URL]
[URL="https://www.jstor.org/stable/2007516?origin=crossref"]https://www.jstor.org/stable/2007516?origin=crossref[/URL]
[URL="http://www.iakovlev.org/zip/riesel2.pdf"]http://www.iakovlev.org/zip/riesel2.pdf[/URL]
[URL="https://www.jstor.org/stable/2006013?origin=crossref"]https://www.jstor.org/stable/2006013?origin=crossref[/URL]
[URL="https://www.jstor.org/stable/2006407?origin=crossref"]https://www.jstor.org/stable/2006407?origin=crossref[/URL]
[URL="https://www.jstor.org/stable/2007382?origin=crossref"]https://www.jstor.org/stable/2007382?origin=crossref[/URL]
[URL="https://www.jstor.org/stable/2033065?origin=crossref"]https://www.jstor.org/stable/2033065?origin=crossref[/URL]
[URL="https://oeis.org/A040076"]https://oeis.org/A040076[/URL]
[URL="https://oeis.org/A046067"]https://oeis.org/A046067[/URL]
[URL="https://oeis.org/A078680"]https://oeis.org/A078680[/URL]
[URL="https://oeis.org/A033809"]https://oeis.org/A033809[/URL]
[URL="https://oeis.org/A040081"]https://oeis.org/A040081[/URL]
[URL="https://oeis.org/A046069"]https://oeis.org/A046069[/URL]
[URL="https://oeis.org/A050412"]https://oeis.org/A050412[/URL]
[URL="https://oeis.org/A108129"]https://oeis.org/A108129[/URL]
[URL="https://oeis.org/A194591"]https://oeis.org/A194591[/URL]
[URL="https://oeis.org/A194636"]https://oeis.org/A194636[/URL]
[URL="https://oeis.org/A050921"]https://oeis.org/A050921[/URL]
[URL="https://oeis.org/A057025"]https://oeis.org/A057025[/URL]
[URL="https://oeis.org/A078683"]https://oeis.org/A078683[/URL]
[URL="https://oeis.org/A038699"]https://oeis.org/A038699[/URL]
[URL="https://oeis.org/A057026"]https://oeis.org/A057026[/URL]
[URL="https://oeis.org/A052333"]https://oeis.org/A052333[/URL]
[URL="https://oeis.org/A291437"]https://oeis.org/A291437[/URL]
[URL="https://oeis.org/A291438"]https://oeis.org/A291438[/URL]
[URL="https://oeis.org/A177330"]https://oeis.org/A177330[/URL]
[URL="https://oeis.org/A345698"]https://oeis.org/A345698[/URL]
[URL="https://oeis.org/A345403"]https://oeis.org/A345403[/URL]
[URL="https://oeis.org/A250204"]https://oeis.org/A250204[/URL]
[URL="https://oeis.org/A250205"]https://oeis.org/A250205[/URL]
[URL="https://oeis.org/A217377"]https://oeis.org/A217377[/URL]
[URL="https://mersenneforum.org/forumdisplay.php?f=81"]https://mersenneforum.org/forumdisplay.php?f=81[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm"]http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm"]http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm"]http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm"]http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm"]http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm[/URL]
[URL="https://www.mersenneforum.org/attachment.php?attachmentid=2966&d=1228059883"]https://www.mersenneforum.org/attachment.php?attachmentid=2966&d=1228059883[/URL]
[URL="http://www.rieselprime.de/Related/LiskovetsGallot.htm"]http://www.rieselprime.de/Related/LiskovetsGallot.htm[/URL]
[URL="https://www.primepuzzles.net/problems/prob_036.htm"]https://www.primepuzzles.net/problems/prob_036.htm[/URL]
[URL="https://oeis.org/A076336/a076336a.html"]https://oeis.org/A076336/a076336a.html[/URL]
[URL="https://oeis.org/A076336/a076336b.html"]https://oeis.org/A076336/a076336b.html[/URL]
[URL="https://www.primepuzzles.net/problems/prob_029.htm"]https://www.primepuzzles.net/problems/prob_029.htm[/URL]
[URL="https://www.rieselprime.de/Related/RieselTwinSG.htm"]https://www.rieselprime.de/Related/RieselTwinSG.htm[/URL]
[URL="http://www.noprimeleftbehind.net/gary/twins100K.htm"]http://www.noprimeleftbehind.net/gary/twins100K.htm[/URL]
[URL="http://www.noprimeleftbehind.net/gary/twins1M.htm"]http://www.noprimeleftbehind.net/gary/twins1M.htm[/URL]
[URL="https://www.primepuzzles.net/problems/prob_049.htm"]https://www.primepuzzles.net/problems/prob_049.htm[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=1647"]http://www.primegrid.com/forum_thread.php?id=1647[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=1731"]http://www.primegrid.com/forum_thread.php?id=1731[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=3980"]http://www.primegrid.com/forum_thread.php?id=3980[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=972"]http://www.primegrid.com/forum_thread.php?id=972[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=1750"]http://www.primegrid.com/forum_thread.php?id=1750[/URL]
[URL="http://www.primegrid.com/forum_thread.php?id=5087"]http://www.primegrid.com/forum_thread.php?id=5087[/URL]
[URL="http://www.primegrid.com/stats_sob_llr.php"]http://www.primegrid.com/stats_sob_llr.php[/URL]
[URL="http://www.primegrid.com/stats_trp_llr.php"]http://www.primegrid.com/stats_trp_llr.php[/URL]
[URL="http://www.primegrid.com/stats_genefer.php"]http://www.primegrid.com/stats_genefer.php[/URL]
[URL="http://www.primegrid.com/stats_psp_llr.php"]http://www.primegrid.com/stats_psp_llr.php[/URL]
[URL="http://www.primegrid.com/stats_esp_llr.php"]http://www.primegrid.com/stats_esp_llr.php[/URL]
[URL="http://www.primegrid.com/stats_sr5_llr.php"]http://www.primegrid.com/stats_sr5_llr.php[/URL]
[URL="http://www.primegrid.com/primes/mega_primes.php"]http://www.primegrid.com/primes/mega_primes.php[/URL]
[URL="https://www.utm.edu/staff/caldwell/preprints/2to100.pdf"]https://www.utm.edu/staff/caldwell/preprints/2to100.pdf[/URL]
[URL="http://www.bitman.name/math/article/1259"]http://www.bitman.name/math/article/1259[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats/all_ck_sierpinski.txt"]http://www.noprimeleftbehind.net/crus/vstats/all_ck_sierpinski.txt[/URL]
[URL="http://www.noprimeleftbehind.net/crus/vstats/all_ck_riesel.txt"]http://www.noprimeleftbehind.net/crus/vstats/all_ck_riesel.txt[/URL]
[URL="https://www.rieselprime.de/Others/CRUS_tab.htm"]https://www.rieselprime.de/Others/CRUS_tab.htm[/URL]
[URL="https://mersenneforum.org/showthread.php?t=11061"]https://mersenneforum.org/showthread.php?t=11061[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=2966&d=1228059883"]https://mersenneforum.org/attachment.php?attachmentid=2966&d=1228059883[/URL]
[URL="https://www.jstor.org/stable/2007898?origin=crossref"]https://www.jstor.org/stable/2007898?origin=crossref[/URL]
[URL="https://oeis.org/A123159"]https://oeis.org/A123159[/URL]
[URL="https://oeis.org/A273987"]https://oeis.org/A273987[/URL]
[URL="https://oeis.org/A283619"]https://oeis.org/A283619[/URL]
[URL="https://oeis.org/A123159/a123159_2.txt"]https://oeis.org/A123159/a123159_2.txt[/URL]
[URL="https://oeis.org/A123159/a123159_1.txt"]https://oeis.org/A123159/a123159_1.txt[/URL]
[URL="https://oeis.org/A123159/a123159.txt"]https://oeis.org/A123159/a123159.txt[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=6277&d=1298454469"]https://mersenneforum.org/attachment.php?attachmentid=6277&d=1298454469[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=6485&d=1303041054"]https://mersenneforum.org/attachment.php?attachmentid=6485&d=1303041054[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519"]https://mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519[/URL]
[URL="https://mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519"]https://mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519[/URL]
[URL="https://en.wikiversity.org/wiki/Sierpinski_problem"]https://en.wikiversity.org/wiki/Sierpinski_problem[/URL] (broken link: [URL="https://web.archive.org/web/20211117071241/https://en.wikiversity.org/wiki/Sierpinski_problem"]from wayback machine[/URL])
[URL="https://en.wikiversity.org/wiki/Riesel_problem"]https://en.wikiversity.org/wiki/Riesel_problem[/URL] (broken link: [URL="https://web.archive.org/web/20211117071111/https://en.wikiversity.org/wiki/Riesel_problem"]from wayback machine[/URL])
[URL="https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub"]https://docs.google.com/document/d/e/2PACX-1vTTLkSb4eY0H19p109lzHjhc-D56gqD9WxyyfQgx_3IEsm2JuA9-cTi1ysy-ahe7RNmc4b9OKKSpYh0/pub[/URL]
[URL="https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub"]https://docs.google.com/document/d/e/2PACX-1vRIjefeGFY7nLpTYSns3JP-aYWGb-4_manLoe1byWwzKmYEW147JaHaC0SfyHF7mwvK29FgpcOr1XfA/pub[/URL]
[URL="https://docs.google.com/document/d/e/2PACX-1vRVGktJ4Vf5qnjManc-oxESDvurmjdhDnyhlUY758r0ZeCkU8wBHl673-6QGKXPQag7peTFYDuiZS7g/pub"]https://docs.google.com/document/d/e/2PACX-1vRVGktJ4Vf5qnjManc-oxESDvurmjdhDnyhlUY758r0ZeCkU8wBHl673-6QGKXPQag7peTFYDuiZS7g/pub[/URL]
[URL="https://docs.google.com/document/d/e/2PACX-1vRsZJeVL5kiS3OgWYgranoCe-SN8L1lF-mGkXjzd0BthR2PXklJ4OG5v2lupBCeAwyn1uBtKeLncFwD/pub"]https://docs.google.com/document/d/e/2PACX-1vRsZJeVL5kiS3OgWYgranoCe-SN8L1lF-mGkXjzd0BthR2PXklJ4OG5v2lupBCeAwyn1uBtKeLncFwD/pub[/URL]
[URL="https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt"]https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Sierpinski%20CK%20for%20bases%20up%20to%202048.txt[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Riesel%20CK%20for%20bases%20up%20to%202048.txt"]https://raw.githubusercontent.com/xayahrainie4793/Extended-Sierpinski-Riesel-conjectures/master/Riesel%20CK%20for%20bases%20up%20to%202048.txt[/URL]
[URL="https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures"]https://github.com/xayahrainie4793/first-4-Sierpinski-Riesel-conjectures[/URL]
[URL="https://github.com/xayahrainie4793/all-k-1024"]https://github.com/xayahrainie4793/all-k-1024[/URL]
[URL="https://github.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base"]https://github.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base[/URL]
[URL="https://www.rechenkraft.net/wiki/Five_or_Bust"]https://www.rechenkraft.net/wiki/Five_or_Bust[/URL]
[URL="https://mersenneforum.org/forumdisplay.php?f=86"]https://mersenneforum.org/forumdisplay.php?f=86[/URL]
[URL="http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf"]http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf[/URL]
[URL="https://oeis.org/A076336/a076336c.html"]https://oeis.org/A076336/a076336c.html[/URL]
[URL="https://oeis.org/A076336/a076336d.html"]https://oeis.org/A076336/a076336d.html[/URL]
[URL="http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt"]http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt[/URL]
[URL="https://oeis.org/A137985/a137985.txt"]https://oeis.org/A137985/a137985.txt[/URL]
[URL="https://mersenneforum.org/showthread.php?t=10761"]https://mersenneforum.org/showthread.php?t=10761[/URL]
[URL="https://mersenneforum.org/showthread.php?t=6545"]https://mersenneforum.org/showthread.php?t=6545[/URL]
[URL="https://mersenneforum.org/showthread.php?t=26328"]https://mersenneforum.org/showthread.php?t=26328[/URL]
[URL="https://oeis.org/A067760"]https://oeis.org/A067760[/URL]
[URL="https://oeis.org/A096502"]https://oeis.org/A096502[/URL]
[URL="https://oeis.org/A276417"]https://oeis.org/A276417[/URL]
[URL="https://oeis.org/A252168"]https://oeis.org/A252168[/URL]
[URL="https://oeis.org/A123252"]https://oeis.org/A123252[/URL]
[URL="https://oeis.org/A096822"]https://oeis.org/A096822[/URL]
[URL="https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj"]https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj[/URL]
[URL="https://oeis.org/A263500"]https://oeis.org/A263500[/URL]
[URL="http://www.prothsearch.com/riesel1.html"]http://www.prothsearch.com/riesel1.html[/URL]
[URL="http://www.prothsearch.com/riesel1a.html"]http://www.prothsearch.com/riesel1a.html[/URL]
[URL="http://www.prothsearch.com/riesel1b.html"]http://www.prothsearch.com/riesel1b.html[/URL]
[URL="http://www.prothsearch.com/riesel1c.html"]http://www.prothsearch.com/riesel1c.html[/URL]
[URL="http://www.prothsearch.com/riesel2.html"]http://www.prothsearch.com/riesel2.html[/URL]
[URL="http://www.prothsearch.com/frequencies.html"]http://www.prothsearch.com/frequencies.html[/URL]
[URL="http://www.15k.org/riesellist.html"]http://www.15k.org/riesellist.html[/URL]
[URL="http://www.15k.org/Summary00300.htm"]http://www.15k.org/Summary00300.htm[/URL]
[URL="http://www.15k.org/Summary02000.htm"]http://www.15k.org/Summary02000.htm[/URL]
[URL="http://www.15k.org/Summary04000.htm"]http://www.15k.org/Summary04000.htm[/URL]
[URL="http://www.15k.org/Summary06000.htm"]http://www.15k.org/Summary06000.htm[/URL]
[URL="http://www.15k.org/Summary08000.htm"]http://www.15k.org/Summary08000.htm[/URL]
[URL="https://oeis.org/wiki/User:Eric_Chen"]https://oeis.org/wiki/User:Eric_Chen[/URL]
[URL="http://www.noprimeleftbehind.net/gary/primes-kx2n-1-001.htm"]http://www.noprimeleftbehind.net/gary/primes-kx2n-1-001.htm[/URL]
[URL="http://www.noprimeleftbehind.net/gary/Rieselprimes-ranges.htm"]http://www.noprimeleftbehind.net/gary/Rieselprimes-ranges.htm[/URL]
[URL="http://www.noprimeleftbehind.net/gary/primes-kx10n-1.htm"]http://www.noprimeleftbehind.net/gary/primes-kx10n-1.htm[/URL]
[URL="https://www.rieselprime.de/default.htm"]https://www.rieselprime.de/default.htm[/URL]
[URL="http://www.fermatsearch.org/factors/faclist.php"]http://www.fermatsearch.org/factors/faclist.php[/URL]
[URL="http://www.fermatsearch.org/factors/composite.php"]http://www.fermatsearch.org/factors/composite.php[/URL]
[URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
[URL="http://www.prothsearch.com/GFN03.html"]http://www.prothsearch.com/GFN03.html[/URL]
[URL="http://www.prothsearch.com/GFN05.html"]http://www.prothsearch.com/GFN05.html[/URL]
[URL="http://www.prothsearch.com/GFN06.html"]http://www.prothsearch.com/GFN06.html[/URL]
[URL="http://www.prothsearch.com/GFN07.html"]http://www.prothsearch.com/GFN07.html[/URL]
[URL="http://www.prothsearch.com/GFN10.html"]http://www.prothsearch.com/GFN10.html[/URL]
[URL="http://www.prothsearch.com/GFN11.html"]http://www.prothsearch.com/GFN11.html[/URL]
[URL="http://www.prothsearch.com/GFN12.html"]http://www.prothsearch.com/GFN12.html[/URL]
[URL="http://www.prothsearch.com/GFNfacs.html"]http://www.prothsearch.com/GFNfacs.html[/URL]
[URL="http://www.prothsearch.com/GFNsmall.html"]http://www.prothsearch.com/GFNsmall.html[/URL]
[URL="http://www.prothsearch.com/GFNsrch.txt"]http://www.prothsearch.com/GFNsrch.txt[/URL]
[URL="http://www.prothsearch.com/OriginalGFNs.html"]http://www.prothsearch.com/OriginalGFNs.html[/URL]
[URL="https://math.stackexchange.com/questions/1394160/conjectured-compositeness-tests-for-n-k-cdot-2n-pm-c"]https://math.stackexchange.com/questions/1394160/conjectured-compositeness-tests-for-n-k-cdot-2n-pm-c[/URL]
[URL="https://math.stackexchange.com/questions/1385519/conjectured-compositeness-tests-for-n-bn-pm-b-pm-1?rq=1"]https://math.stackexchange.com/questions/1385519/conjectured-compositeness-tests-for-n-bn-pm-b-pm-1?rq=1[/URL]
[URL="https://math.stackexchange.com/questions/1426586/conjectured-compositeness-tests-for-n-k-cdot-bn-pm-c?rq=1"]https://math.stackexchange.com/questions/1426586/conjectured-compositeness-tests-for-n-k-cdot-bn-pm-c?rq=1[/URL]
[URL="https://oeis.org/A305237"]https://oeis.org/A305237[/URL]
[URL="https://oeis.org/A325204"]https://oeis.org/A325204[/URL]
[URL="https://math.stackexchange.com/questions/3345481/three-consecutive-numbers-with-exactly-different-four-prime-factors"]https://math.stackexchange.com/questions/3345481/three-consecutive-numbers-with-exactly-different-four-prime-factors[/URL]
[URL="https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes"]https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes[/URL]
[URL="https://oeis.org/A055557/a055557.txt"]https://oeis.org/A055557/a055557.txt[/URL]
[URL="https://oeis.org/A007013/a007013.pdf"]https://oeis.org/A007013/a007013.pdf[/URL]
[URL="https://oeis.org/A005165/a005165.pdf"]https://oeis.org/A005165/a005165.pdf[/URL]
[URL="https://www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0480311-0/S0025-5718-1978-0480311-0.pdf"]https://www.ams.org/journals/mcom/1978-32-144/S0025-5718-1978-0480311-0/S0025-5718-1978-0480311-0.pdf[/URL]
[URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL]
[URL="https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf"]https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf[/URL]
[URL="https://cbright.myweb.cs.uwindsor.ca/reports/mepn.pdf"]https://cbright.myweb.cs.uwindsor.ca/reports/mepn.pdf[/URL]
[URL="https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf"]https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf[/URL]
[URL="https://cbright.myweb.cs.uwindsor.ca/talks/minimal-slides.pdf"]https://cbright.myweb.cs.uwindsor.ca/talks/minimal-slides.pdf[/URL]
[URL="https://doi.org/10.1080/10586458.2015.1064048"]https://doi.org/10.1080/10586458.2015.1064048[/URL]
[URL="https://github.com/curtisbright/mepn-data/blob/master/report/report.tex"]https://github.com/curtisbright/mepn-data/blob/master/report/report.tex[/URL]
[URL="http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf"]http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf[/URL] (base 10)
[URL="http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/minimal.pdf"]http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/minimal.pdf[/URL] (base 10)
[URL="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.5686&rep=rep1&type=pdf"]http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.7.5686&rep=rep1&type=pdf[/URL] (base 10)
[URL="https://scholar.colorado.edu/downloads/hh63sw661"]https://scholar.colorado.edu/downloads/hh63sw661[/URL] (bases <= 10)
[URL="https://cbright.myweb.cs.uwindsor.ca/reports/cs662-problem12.pdf"]https://cbright.myweb.cs.uwindsor.ca/reports/cs662-problem12.pdf[/URL] (base 13, family 8{0}111)
[URL="https://oeis.org/A347819/a347819.pdf"]https://oeis.org/A347819/a347819.pdf[/URL]
[URL="https://cs.uwaterloo.ca/~shallit/papers.html"]https://cs.uwaterloo.ca/~shallit/papers.html[/URL]
[URL="http://www.curtisbright.com/"]http://www.curtisbright.com/[/URL]
[URL="https://cbright.myweb.cs.uwindsor.ca/"]https://cbright.myweb.cs.uwindsor.ca/[/URL]
[URL="https://www.researchgate.net/profile/Curtis-Bright"]https://www.researchgate.net/profile/Curtis-Bright[/URL]
[URL="http://www.bitman.name/math/article/730"]http://www.bitman.name/math/article/730[/URL]
[URL="https://github.com/curtisbright/mepn-data"]https://github.com/curtisbright/mepn-data[/URL]
[URL="https://github.com/curtisbright/mepn-data/archive/refs/heads/master.zip"]https://github.com/curtisbright/mepn-data/archive/refs/heads/master.zip[/URL]
[URL="https://github.com/RaymondDevillers/primes"]https://github.com/RaymondDevillers/primes[/URL]
[URL="https://github.com/RaymondDevillers/primes/archive/refs/heads/master.zip"]https://github.com/RaymondDevillers/primes/archive/refs/heads/master.zip[/URL]
[URL="https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes"]https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes[/URL]
[URL="https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes/archive/refs/heads/master.zip"]https://github.com/xayahrainie4793/minimal-primes-and-left-right-truncatable-primes/archive/refs/heads/master.zip[/URL]
[URL="https://github.com/xayahrainie4793/non-single-digit-primes"]https://github.com/xayahrainie4793/non-single-digit-primes[/URL]
[URL="https://github.com/xayahrainie4793/non-single-digit-primes/archive/refs/heads/main.zip"]https://github.com/xayahrainie4793/non-single-digit-primes/archive/refs/heads/main.zip[/URL]
[URL="https://github.com/xayahrainie4793/mepn/tree/primes-greater-than-base"]https://github.com/xayahrainie4793/mepn/tree/primes-greater-than-base[/URL]
[URL="https://github.com/xayahrainie4793/mepn/archive/refs/heads/primes-greater-than-base.zip"]https://github.com/xayahrainie4793/mepn/archive/refs/heads/primes-greater-than-base.zip[/URL]
[URL="http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html"]http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html[/URL]
[URL="http://www.bitman.name/math/table/497"]http://www.bitman.name/math/table/497[/URL]
[URL="http://www.bitman.name/math/table/498"]http://www.bitman.name/math/table/498[/URL]
[URL="http://www.bitman.name/math/table/499"]http://www.bitman.name/math/table/499[/URL]
[URL="http://www.bitman.name/math/table/500"]http://www.bitman.name/math/table/500[/URL]
[URL="http://www.bitman.name/math/table/501"]http://www.bitman.name/math/table/501[/URL]
[URL="http://www.bitman.name/math/table/504"]http://www.bitman.name/math/table/504[/URL]
[URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub[/URL] (new link from GoogleDrive: [URL="https://docs.google.com/document/d/17RtAuTOGMJOYjbyf24zcPJqEHQQQwC_1A6EN154rpFs/edit?usp=sharing"]https://docs.google.com/document/d/17RtAuTOGMJOYjbyf24zcPJqEHQQQwC_1A6EN154rpFs/edit?usp=sharing[/URL])
[URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]
[URL="https://www.primepuzzles.net/puzzles/puzz_178.htm"]https://www.primepuzzles.net/puzzles/puzz_178.htm[/URL]
[URL="https://www.primepuzzles.net/problems/prob_083.htm"]https://www.primepuzzles.net/problems/prob_083.htm[/URL]
[URL="http://recursed.blogspot.com/2006/12/prime-game.html"]http://recursed.blogspot.com/2006/12/prime-game.html[/URL]
[URL="http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf"]http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/primes2.pdf[/URL]
[URL="https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it"]https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it[/URL]
[URL="https://cs.stackexchange.com/questions/48084/determining-if-infinite-binary-language-dfas-contain-at-least-1-prime"]https://cs.stackexchange.com/questions/48084/determining-if-infinite-binary-language-dfas-contain-at-least-1-prime[/URL]
[URL="https://logs.esolangs.org/freenode-esoteric/2011-02-04.html"]https://logs.esolangs.org/freenode-esoteric/2011-02-04.html[/URL]
[URL="https://mersenneforum.org/showpost.php?p=562621&postcount=7"]https://mersenneforum.org/showpost.php?p=562621&postcount=7[/URL]
[URL="https://oeis.org/A326609"]https://oeis.org/A326609[/URL]
[URL="https://oeis.org/A330048"]https://oeis.org/A330048[/URL]
[URL="https://oeis.org/A330049"]https://oeis.org/A330049[/URL]
[URL="https://oeis.org/A347819"]https://oeis.org/A347819[/URL]
[URL="https://oeis.org/A071062"]https://oeis.org/A071062[/URL]
[URL="https://oeis.org/A111055"]https://oeis.org/A111055[/URL]
[URL="https://oeis.org/A111056"]https://oeis.org/A111056[/URL]
[URL="https://oeis.org/A114835"]https://oeis.org/A114835[/URL]
[URL="https://oeis.org/A110600"]https://oeis.org/A110600[/URL]
[URL="https://oeis.org/A111057"]https://oeis.org/A111057[/URL]
[URL="https://oeis.org/A071070"]https://oeis.org/A071070[/URL]
[URL="https://oeis.org/A110615"]https://oeis.org/A110615[/URL]

sweety439 2021-02-17 14:32

1 Attachment(s)
I tried to write a PARI/GP code that can print all minimal primes (start with b+1) up to length 1000 in given base b in <10 minutes, but not success, since the code for updating L (when L contains non-simple families) by "let w be the shortest sting in this family, if w has a subword in M, then remove the family from L, if w represents a prime, then add w to M, if the family can be proven to only contain composites, then remove the family from L" (see page of [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL]) is very complex.

Thus, I only have the program that looks the small primes one-by-one, and I only checked the simple families of the from x{y}, x{y}, and x{0}y (where x,y are digits) to find the smallest primes in these simple families (or to prove that these simple families only contain composites).

The more difficult case is: Non-simple families that can be proven to only contain composites, if the gcd (greatest common divisor) of the digits in these families is >1, then these families clearly only contain composites (note: we only count the numbers > base), but there exist many non-simple families with gcd of the digits = 1 and can be proven to only contain composites (and all subsequences of all numbers in these families represent composites, when we only count the numbers > base), e.g.

{1}6{1} in base 9
{3}{0}5 in base 9
{3}{6}8 in base 9

(base 9 is the first base which has such families)

Interestingly, base 9 is also the first base with some simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), since base b has such families x{y} or {x}y if and only if b+1 is not prime or prime power, and gcd(repeating digit, b+1) = 1, the first base b such that b+1 is not prime or prime power is 5, but for base 5, the only such families are 3{1}, 4{1}, {1}3, {1}4, but the smallest prime in the family whose repeating digit is 1 may not be minimal prime (start with b+1), unless base b has no repunit primes (the first such bases b are 9, 25, 32, 49, 64, ...), and base 5 has repunit prime 111 (=31 in base 10), thus base 5 has no simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), and next base b such that b+1 is not prime or prime power is 9, and base 9 has these simple families: 2{7}, 5{1}, 5{7}, 6{1}, {7}2, {1}5, {3}5, {7}5, {3}8, which are ruled out as only contain composites by covering set {2,5} (also the families 5{3}, 8{3}, {1}6, but they are already ruled out as only contain composites by trivial 1-cover set {3})

Since in any base b, for a repdigit (a number whose all digits are all same) to be prime (only count numbers > base), it must be a repunit and have a prime number of digits in its base (b), and for the simple families x{y} and {x}y in base b, the only chance of their smallest primes (if exist) are not minimal primes (start with b+1) in base b is the base b repdigit is prime, thus the repeating digit in these families must be 1, and since in bases 9, 25, 32, 49, 64, ... there are no repunit primes, thus in these bases, the smallest primes (if exist) in all simple families x{y} and {x}y are [I]always[/I] minimal primes (start with b+1) in base b

sweety439 2021-02-17 14:36

Except the families *{0}1 and *{z} (where * represents any string of digits), when the corresponding prime is large, the known [URL="https://primes.utm.edu/prove/index.html"]primality tests[/URL] for such a number are too inefficient to run (*{0}1 can be proven prime by [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 primality test[/URL], *{z} can be proven prime by [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 primality test[/URL]). In this case one must resort to a [URL="https://primes.utm.edu/glossary/page.php?sort=PRP"]probable primality[/URL] test such as [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL] and [URL="https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html"]Baillie–PSW primality test[/URL], unless a divisor of the number can be found, since we are testing many numbers in an exponential sequence, it is possible to use a sieving process (such as [URL="http://www.rieselprime.de/dl/CRUS_pack.zip"]srsieve[/URL] software) to find divisors rather than using [URL="https://primes.utm.edu/glossary/page.php?sort=TrialDivision"]trial division[/URL]. (since the only known primality tests with factorization a number are [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 test[/URL] and [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 test[/URL], and the only families which can be trivially 100% factored are *{0} (where * represents any string of digits) (trivially factored to product of * and b^n (where n is the number of 0's in the bracket)), thus N-1 or N+1 must be *{0}, and N-1 is *{0} only when N is *{0}1, and N+1 is *{0} only when N is *{z})

There are three levels for these large minimal (probable) primes (start with b+1) base b: (let the large minimal (probable) primes (start with b+1) base b be N, and assume N > 10^3000, since all (probable) primes < 10^3000 can be easily proven prime by [URL="http://www.ellipsa.eu/public/primo/primo.html"]Primo[/URL])

1. either N-1 or N+1 can be trivially 100% factored (i.e. the primes in families *{0}1 and *{z} (where * represents any string of digits) in any base) (e.g. [URL="http://factordb.com/index.php?id=1100000000765961441"]39(0^6266)1 base 13[/URL], [URL="http://factordb.com/index.php?id=1100000000884560233"]4(D^19698) base 14[/URL], [URL="http://factordb.com/index.php?id=1100000000034167087"]A(0^1355)1 base 17[/URL], [URL="http://factordb.com/index.php?id=1100000000885544949"]F1(0^18523)1 base 19[/URL], [URL="http://factordb.com/index.php?id=1100000000777265872"]5D(0^19848)1 base 21[/URL], [URL="http://factordb.com/index.php?id=1100000000212668509"]4(0^341)1 base 23[/URL], [URL="http://factordb.com/index.php?id=1100000000634720609"]8(0^119214)1 base 23[/URL], [URL="http://factordb.com/index.php?id=1100000000785448736"]C(0^1022)1 base 30[/URL], [URL="http://factordb.com/index.php?id=1100000000800812865"]O(T^34205) base 30[/URL], [URL="http://factordb.com/index.php?id=1100000000817923446"]13(0^23614)1 base 33[/URL], [URL="http://factordb.com/index.php?id=1100000000838755581"]N7(0^610411)1 base 33[/URL], [URL="http://factordb.com/index.php?id=1100000000904766458"]Q(X^3086) base 34[/URL], [URL="http://factordb.com/index.php?id=1100000000885460611"]1B(0^56061)1 base 35[/URL], [URL="http://factordb.com/index.php?id=1100000000838600210"]FY(a^22021) base 37[/URL], [URL="http://factordb.com/index.php?id=1100000000838600120"]R8(a^20895) base 37[/URL], [URL="http://factordb.com/index.php?id=1100000000765960286"]2(0^2728)1 base 38[/URL], [URL="http://factordb.com/index.php?id=1100000001533872954"]V(0^1527)1 base 38[/URL], [URL="http://factordb.com/index.php?id=1100000000904762826"]L(b^1579) base 38[/URL], [URL="http://factordb.com/index.php?id=1100000000836244858"]a(b^136211) base 38[/URL], [URL="http://factordb.com/index.php?id=1100000000784120237"]2(f^2523) base 42[/URL], [URL="http://factordb.com/index.php?id=1100000000819408168"]N(i^153355) base 45[/URL], [URL="http://factordb.com/index.php?id=1100000000885544904"]O(0^18521)1 base 45[/URL], [URL="http://factordb.com/index.php?id=1100000000767042575"]3(k^1555) base 47[/URL], [URL="http://factordb.com/index.php?id=1100000001059907862"]T(0^133041)1 base 48[/URL], [URL="http://factordb.com/index.php?id=1100000000765961712"]7(0^515)1 base 50[/URL], [URL="http://factordb.com/index.php?id=1100000001538081180"]c(0^4880)1 base 51[/URL], [URL="http://factordb.com/index.php?id=1100000002311482172"]g(0^4821)1 base 52[/URL], [URL="http://factordb.com/index.php?id=1100000000942823223"]8(0^227182)1 base 53[/URL], [URL="http://factordb.com/index.php?id=1100000000809680743"]E(0^14954)1 base 57[/URL], [URL="http://factordb.com/index.php?id=1100000000935833895"]L(0^1030)1 base 58[/URL], [URL="http://factordb.com/index.php?id=1100000000413676848"]N($^3020) base 64[/URL]), in this case we can use either Pocklington [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 method[/URL] or Morrison [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 method[/URL] to prove the primility of this minimal prime (start with b+1) base b.

2. neither N-1 nor N+1 can be trivially 100% factored, but either N-1 or N+1 can be trivially factored to product to a small number and a large base b repunit number (e.g. the generalized repunit primes (the primes in family {1}) (since {1} - 1 = {1}0 and {1}0 = 10 (small number) * {1} (repunit number)), the generalized half Fermat primes (the primes in family {#}$, for odd base b, # = (b-1)/2, $ = (b+1)/2) (since {#}$ - 1 = {#} and {#} = # (small number) * {1} (repunit number)), and the primes in families 1{2} (since 1{2} - 1 = 1{2}1 and 1{2}1 = 11 (small number) * {1} (repunit number)), 1{3} (since 1{3} - 1 = 1{3}2 and 1{3}2 = 12 (small number) * {1} (repunit number)), 1{4} (since 1{4} - 1 = 1{4}3 and 1{4}3 = 13 (small number) * {1} (repunit number)), {2}1 (since {2}1 - 1 = {2}0 and {2}0 = 20 (small number) * {1} (repunit number), also {2}1 + 1 = {2} and {2} = 2 (small number) * {1} (repunit number)), {3}1 (since {3}1 - 1 = {3}0 and {3}0 = 30 (small number) * {1} (repunit number)), {4}1 (since {4}1 - 1 = {4}0 and {4}0 = 40 (small number) * {1} (repunit number)), {2}3 (since {2}3 - 1 = {2} and {2} = 2 (small number) * {1} (repunit number)), {3}2 (since {3}2 + 1 = {3} and {3} = 3 (small number) * {1} (repunit number)), {3}4 (since {3}4 - 1 = {3} and {3} = 3 (small number) * {1} (repunit number)), {4}3 (since {4}3 + 1 = {4} and {4} = 4 (small number) * {1} (repunit number)), in any base) (e.g. [URL="http://factordb.com/index.php?id=1100000002321021456"]1(B^576) base 13[/URL], [URL="http://factordb.com/index.php?id=1100000002320890755"](7^1504)1 base 13[/URL], [URL="http://factordb.com/index.php?id=1100000000840126705"](9^308)1 base 13[/URL], [URL="http://factordb.com/index.php?id=1100000000000217927"](B^563)C base 13[/URL], [URL="http://factordb.com/index.php?id=1100000000840355814"](9^292)1 base 17[/URL], [URL="http://factordb.com/index.php?id=1100000000840383833"](G^2034)1 base 19[/URL], [URL="http://factordb.com/index.php?id=1100000002325396014"](3^1063)2 base 21[/URL], [URL="http://factordb.com/index.php?id=1100000002325398836"](7^230)1 base 21[/URL], [URL="http://factordb.com/index.php?id=1100000001603032659"](F^1091)G base 23[/URL], [URL="http://factordb.com/index.php?id=1100000002326031108"](H^1020)1 base 23[/URL], [URL="http://factordb.com/index.php?id=1100000000934823810"](K^3761)L base 23[/URL], [URL="http://factordb.com/index.php?id=1100000002611727808"](B^305)C base 25[/URL], [URL="http://factordb.com/index.php?id=1100000000840632517"](8^354)1 base 26[/URL], [URL="http://factordb.com/index.php?id=1100000002328031251"]1(H^4272) base 27[/URL], [URL="http://factordb.com/index.php?id=1100000002327660879"](2^1986)1 base 31[/URL], [URL="http://factordb.com/index.php?id=1100000002327662885"](3^4260)1 base 31[/URL], [URL="http://factordb.com/index.php?id=1100000000903613914"](P^1025)Q base 31[/URL], [URL="http://factordb.com/index.php?id=1100000000899426975"](V^251)W base 33[/URL], [URL="http://factordb.com/index.php?id=1100000000012776520"](1^313) base 35[/URL], [URL="http://factordb.com/index.php?id=1100000000012789513"](1^349) base 39[/URL], [URL="http://factordb.com/index.php?id=1100000000467236538"](1^4229) base 51[/URL]) or can be factored to product to a small number and b^n+1 with large n (e.g. [URL="http://factordb.com/index.php?id=1100000000633424191"]9(0^3542)91 base 16[/URL], [URL="http://factordb.com/index.php?id=1100000000840383633"]F(0^293)E base 19[/URL], [URL="http://factordb.com/index.php?id=1100000002355608589"]B(0^3529)C base 25[/URL], [URL="http://factordb.com/index.php?id=1100000002356302186"]C(0^544)D base 29[/URL]), in this case we require the factored part at least 33.333% for the (base b) repunit number, and the base b repunit number with length n has algebra factors: Phi_d(b) (where Phi is [URL="https://en.wikipedia.org/wiki/Cyclotomic_polynomial"]cyclotomic polynomial[/URL]) for all d>1 dividing n), thus these numbers can be proven prime if these Phi_d(b) can be factored to make N-1 or N+1 over 33.333% factored, and this is equivalent to factor the Cunningham numbers b^n+-1 (references for factoring Cunningham numbers: [URL="https://homes.cerias.purdue.edu/~ssw/cun/index.html"]b<=12[/URL] [URL="https://maths-people.anu.edu.au/~brent/pub/pub134.html"]13<=b<=99[/URL] [URL="https://stdkmd.net/nrr/repunit/"]b=10[/URL] [URL="http://myfactors.mooo.com/"]any b[/URL]), if this base b repunit number at least 33.333% factored part, then we can prove the primility for this minimal prime (start with b+1) base b, otherwise we can only use [URL="https://primes.utm.edu/glossary/page.php?sort=PRP"]probable primality[/URL] test (since the known [URL="https://primes.utm.edu/prove/index.html"]primality tests[/URL] for such a number are too inefficient to run) such as [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL] and [URL="https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html"]Baillie–PSW primality test[/URL] to show that this number is probable prime, and the possibility of this number is in fact composite is less than 10^(-679) if this minimal prime (start with b+1) base b is larger than 10^5000, reference: [URL="https://primes.utm.edu/notes/prp_prob.html"]https://primes.utm.edu/notes/prp_prob.html[/URL]

3. neither N-1 nor N+1 can be trivially 100% factored or trivially factored to product to a small number and a large base b repunit number, in this case we can only use [URL="https://primes.utm.edu/glossary/page.php?sort=PRP"]probable primality[/URL] test (since the known [URL="https://primes.utm.edu/prove/index.html"]primality tests[/URL] for such a number are too inefficient to run) such as [URL="https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test"]Miller–Rabin primality test[/URL] and [URL="https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html"]Baillie–PSW primality test[/URL] to show that this number is probable prime, and the possibility of this number is in fact composite is less than 10^(-679) if this minimal prime (start with b+1) base b is larger than 10^5000, reference: [URL="https://primes.utm.edu/notes/prp_prob.html"]https://primes.utm.edu/notes/prp_prob.html[/URL]

however, in some primes which are case 2 or case 3, N-1 or N+1 still has algebra factors (like that some [URL="https://brnikat.com/nums/cullen_woodall/algebraic.txt"]generalized Cullen/Woodall numbers[/URL] have algebra factors) to make it over 33.333% factored, such as difference-of-squares factorization or difference-of-cubes factorization, e.g. [URL="http://factordb.com/index.php?id=1100000002355574745"]8(0^298)B base 18[/URL], N+1 = 8*18^299+12 = (18^2)*(8*18^297)+12 = 12*27*(8*18^297)+12 = 12*(27*(8*18^297)+1) = 12*(3*(2*18^99)+1)*(9*(4*18^198)-3*(2*18^99)+1) has sum-of-cubes factorization, to make it over 33.333% factored and thus this number can be proven prime with N+1 method, a non-example is [URL="http://factordb.com/index.php?id=1100000002355610241"]2(0^313)7 base 24[/URL], N+1 = 2*24^314+8 = 2^943*3^314+8 = 8*(2^940*3^314+1), N-1 = 2*24^314+6 = 2*(24^314+3) = 2*(2^942*3^314+3) = 6*(2^941*3^313+1), neither of them has algebra factorization, thus we can do nothing but using Primo to prove its primality.

[URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]examples of prove the primility for the generalized repunit primes by factoring Phi_d(b) for d dividing n, click the link of the numbers in "Prime for Exponent" column[/URL]

sweety439 2021-02-17 14:42

If the [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski[/URL]/[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel[/URL] CK for base b is <b (see [URL="http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm"]http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm[/URL] for the list of the CK for bases 2<=b<=1030), then the "minimal primes (start with b+1) base b problem" covers the Sierpinski base b problem and the Riesel base b problem, since [I]all[/I] primes for the Sierpinski base b problem and the Riesel base b problem are minimal primes (start with b+1) base b

sweety439 2021-02-18 11:42

There are no base b and simple family x{y}z (where x, y, z are base b digit strings) such that all numbers in this family (i.e. xz, xyz, xyyz, xyyyz, ... in base b) are primes, like that there are no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n, see [URL="https://en.wikipedia.org/wiki/Formula_for_primes#Prime_formulas_and_polynomial_functions"]https://en.wikipedia.org/wiki/Formula_for_primes#Prime_formulas_and_polynomial_functions[/URL] (hence there are no [URL="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression"]primes in arithmetic progression[/URL] with infinite length), and there are no (1st kind or 2nd kind) [URL="https://en.wikipedia.org/wiki/Cunningham_chain"]Cunningham chains[/URL] with infinite length.

sweety439 2021-02-18 11:46

1 Attachment(s)
The length of the minimal primes (start with b+1) in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 30 are:

(exceed Mersenneforum's 65536 characters limit, thus upload text file)

They appear to follow the [URL="https://en.wikipedia.org/wiki/Benford%27s_law"]Benford's law[/URL], i.e. for n>=2, the number of n-digit minimal primes (start with b+1) base b is inversely proportional to n-1 (i.e. the expected value is c/(n-1), where c is a fixed constant) (i.e. the graph of the points (x,y=number of x-digit minimal primes (start with b+1)) in the xy-plane is near to the graph of y=c/(x-1) in the xy-plane for a fixed real number c), for any fixed base b

VBCurtis 2021-02-18 15:31

Your work does not rate daily posts to update us.

If you continue to post to this thread every single day, you're going to find yourself with time off again.

Try monthly update posts. Yes, monthly.

You can edit your previously posted attachments without triggering a new-post notice to all the mods- try that too. But if you keep drawing attention to your endless procession of trivial update posts, you're likely to lose the ability to make those posts.

sweety439 2021-02-18 15:42

The largest possible appearance for given digit d in minimal prime (start with b+1) in base b:

If base b has repunit primes, then the largest possible appearance for digit d=1 in minimal prime (start with b+1) in base b is the length of smallest repunit prime base b (i.e. [URL="https://oeis.org/A084740"]A084740[/URL](b)), the first bases which do not have repunit primes are 9, 25, 32, 49, 64, ...

[CODE]
b=2, d=0: 0
b=2, d=1: 2 (the prime 11)
b=3, d=0: 0
b=3, d=1: 3 (the prime 111)
b=3, d=2: 1 (the primes 12 and 21)
b=4, d=0: 0
b=4, d=1: 2 (the prime 11)
b=4, d=2: 2 (the prime 221)
b=4, d=3: 1 (the primes 13, 23, 31)
b=5, d=0: 93 (the prime 10[SUB]93[/SUB]13)
b=5, d=1: 3 (the prime 111)
b=5, d=2: 1 (the primes 12, 21, 23, 32)
b=5, d=3: 4 (the prime 33331)
b=5, d=4: 4 (the primes 14444 and 44441)
b=6, d=0: 2 (the prime 40041)
b=6, d=1: 2 (the prime 11)
b=6, d=2: 1 (the primes 21 and 25)
b=6, d=3: 1 (the primes 31 and 35)
b=6, d=4: 3 (the prime 4441)
b=6, d=5: 1 (the primes 15, 25, 35, 45, 51)
b=7, d=0: 7 (the prime 5100000001)
b=7, d=1: 5 (the prime 11111)
b=7, d=2: 3 (the prime 1222)
b=7, d=3: 16 (the prime 3[SUB]16[/SUB]1)
b=7, d=4: 2 (the primes 344, 445, 544, 4504, 40054)
b=7, d=5: 4 (the prime 35555)
b=7, d=6: 2 (the prime 6634)
b=8, d=0: 3 (the prime 500025)
b=8, d=1: 3 (the prime 111)
b=8, d=2: 2 (the prime 225)
b=8, d=3: 3 (the prime 3331)
b=8, d=4: 220 (the prime 4[SUB]220[/SUB]7)
b=8, d=5: 14 (the prime 5[SUB]13[/SUB]25)
b=8, d=6: 2 (the primes 661 and 667)
b=8, d=7: 12 (the prime 7[SUB]12[/SUB]1)
b=9, d=0: 1158 (the prime 30[SUB]1158[/SUB]11)
b=9, d=1: 36 (the prime 561[SUB]36[/SUB])
b=9, d=2: 4 (the prime 22227)
b=9, d=3: 8 (the prime 8333333335)
b=9, d=4: 11 (the prime 54[SUB]11[/SUB])
b=9, d=5: 4 (the prime 55551)
b=9, d=6: 329 (the prime 76[SUB]329[/SUB]2)
b=9, d=7: 687 (the prime 27[SUB]686[/SUB]07)
b=9, d=8: 19 (the prime 8[SUB]19[/SUB]335)
b=10, d=0: 28 (the prime 50[SUB]28[/SUB]27)
b=10, d=1: 2 (the prime 11)
b=10, d=2: 3 (the prime 2221)
b=10, d=3: 1 (the primes 13, 23, 31, 37, 43, 53, 73, 83, 349)
b=10, d=4: 2 (the prime 449)
b=10, d=5: 11 (the prime 5[SUB]11[/SUB]1)
b=10, d=6: 4 (the prime 666649)
b=10, d=7: 2 (the primes 277, 577, 727, 757, 787, 877)
b=10, d=8: 2 (the prime 881)
b=10, d=9: 3 (the prime 9949)
b=11, d=0: 126 (the prime 50[SUB]126[/SUB]57)
b=11, d=1: 17 (the prime 1[SUB]17[/SUB])
b=11, d=2: 6 (the prime 5222222)
b=11, d=3: 10 (the prime 3[SUB]10[/SUB]7)
b=11, d=4: 44 (the prime 4[SUB]44[/SUB]1)
b=11, d=5: 221 (the prime 85[SUB]220[/SUB]05]
b=11, d=6: 124 (the prime 326[SUB]124[/SUB])
b=11, d=7: 62668 (the prime 57[SUB]62668[/SUB])
b=11, d=8: 17 (the prime 8[SUB]17[/SUB]3)
b=11, d=9: 32 (the prime 9[SUB]32[/SUB]1)
b=11, d=A: 713 (the prime A[SUB]713[/SUB]58)
b=12, d=0: 39 (the prime 40[SUB]39[/SUB]77)
b=12, d=1: 2 (the prime 11)
b=12, d=2: 3 (the prime 222B)
b=12, d=3: 1 (the primes 31, 35, 37, 3B)
b=12, d=4: 3 (the prime 4441)
b=12, d=5: 2 (the primes 565 and 655)
b=12, d=6: 2 (the prime 665)
b=12, d=7: 3 (the primes 4777 and 9777)
b=12, d=8: 1 (the primes 81, 85, 87, 8B)
b=12, d=9: 4 (the prime 9999B)
b=12, d=A: 4 (the prime AAAA1)
b=12, d=B: 7 (the prime BBBBBB99B)
b=13, d=0: 32017 (the prime 80[SUB]32017[/SUB]111)
b=13, d=1: 5 (the prime 11111)
b=13, d=2: 77 (the prime 72[SUB]77[/SUB])
b=13, d=3: >82000 (the prime A3[SUB]n[/SUB]A)
b=13, d=4: 14 (the prime 94[SUB]14[/SUB])
b=13, d=5: >88000 (the prime 95[SUB]n[/SUB])
b=13, d=6: 137 (the prime 6[SUB]137[/SUB]A3)
b=13, d=7: 1504 (the prime 7[SUB]1504[/SUB]1)
b=13, d=8: 53 (the prime 8[SUB]53[/SUB]7)
b=13, d=9: 1362 (the prime 9[SUB]1362[/SUB]5)
b=13, d=A: 95 (the prime C5A[SUB]95[/SUB])
b=13, d=B: 834 (the prime B[SUB]834[/SUB]74)
b=13, d=C: 10631 (the prime C[SUB]10631[/SUB]92)
b=14, d=0: 83 (the prime 40[SUB]83[/SUB]49)
b=14, d=1: 3 (the prime 111)
b=14, d=2: 3 (the prime B2225)
b=14, d=3: 5 (the prime A33333)
b=14, d=4: 63 (the prime 4[SUB]63[/SUB]09)
b=14, d=5: 36 (the prime 85[SUB]36[/SUB])
b=14, d=6: 10 (the prime 86[SUB]10[/SUB]99)
b=14, d=7: 2 (the primes 771, 77D)
b=14, d=8: 86 (the prime 8[SUB]86[/SUB]B)
b=14, d=9: 37 (the prime 9[SUB]36[/SUB]89)
b=14, d=A: 59 (the prime A[SUB]59[/SUB]3)
b=14, d=B: 78 (the prime 6B[SUB]77[/SUB]2B)
b=14, d=C: 79 (the prime 8C[SUB]79[/SUB]3)
b=14, d=D: 19698 (the prime 4D[SUB]19698[/SUB])
b=15, d=0: 33 (the prime 50[SUB]33[/SUB]17)
b=15, d=1: 3 (the prime 111)
b=15, d=2: 9 (the prime 2222222252)
b=15, d=3: 12 (the prime 3[SUB]12[/SUB]1)
b=15, d=4: 3 (the prime 4434)
b=15, d=5: 8 (the prime 555555557)
b=15, d=6: 104 (the prime 96[SUB]104[/SUB]08)
b=15, d=7: 156 (the prime 7[SUB]155[/SUB]97)
b=15, d=8: 8 (the prime 8888888834)
b=15, d=9: 10 (the prime 9999999999D)
b=15, d=A: 4 (the prime AAAA52)
b=15, d=B: 31 (the prime EB[SUB]31[/SUB])
b=15, d=C: 10 (the prime DCCCCCCCCCC8)
b=15, d=D: 16 (the prime D[SUB]16[/SUB]B)
b=15, d=E: 145 (the prime E[SUB]145[/SUB]397)
b=16, d=0: 3542 (the prime 90[SUB]3542[/SUB]91)
b=16, d=1: 2 (the prime 11)
b=16, d=2: 32 (the prime 2[SUB]32[/SUB]7)
b=16, d=3: >76000 (the prime 3[SUB]n[/SUB]AF)
b=16, d=4: 72785 (the prime 4[SUB]72785[/SUB]DD)
b=16, d=5: 70 (the prime A015[SUB]70[/SUB])
b=16, d=6: 87 (the prime 56[SUB]87[/SUB]F)
b=16, d=7: 20 (the prime 7[SUB]19[/SUB]87)
b=16, d=8: 1517 (the prime F8[SUB]1517[/SUB]F)
b=16, d=9: 1052 (the prime D9[SUB]1052[/SUB])
b=16, d=A: 305 (the prime DA[SUB]305[/SUB]5)
b=16, d=B: 32234 (the prime DB[SUB]32234[/SUB])
b=16, d=C: 3700 (the prime 5BC[SUB]3700[/SUB]D)
b=16, d=D: 39 (the prime 4D[SUB]39[/SUB])
b=16, d=E: 34 (the prime E[SUB]34[/SUB]B)
b=16, d=F: 1961 (the prime 300F[SUB]1960[/SUB]AF)
[/CODE]

sweety439 2021-06-02 12:06

2 Attachment(s)
Update the text file for all known minimal primes (start with b+1) in bases 2<=b<=16

Note: Only bases 2, 3, 4, 5, 6, 7, 8, 10, 12 are completely solved.

In general, if [URL="https://oeis.org/A000010"]eulerphi[/URL](b) is larger than [URL="https://oeis.org/A000010"]eulerphi[/URL](c), than base b is more difficult than base c, since [URL="https://oeis.org/A000010"]eulerphi[/URL](b) is the number of possible last digit for a prime >b in base b, since a base-b digit can be the last digit for a prime >b in base b if and only if gcd(this digit,b) = 1 (the number of possible first digit for a prime >b in base b is b-1, since all digits except 0 can be the first digit). The difficulty of this problem (finding all minimal primes (start with b+1) in base b) is about [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^[URL="https://oeis.org/A000010"]eulerphi[/URL](b), also, the number of minimal primes (start with b+1) in base b is about [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^[URL="https://oeis.org/A000010"]eulerphi[/URL](b), also, the length of largest minimal prime (start with b+1) in base b is about [URL="https://en.wikipedia.org/wiki/E_(mathematical_constant)"]e[/URL]^([URL="https://oeis.org/A000010"]eulerphi[/URL](b)^2)

[CODE]
level ([URL="https://oeis.org/A000010"]eulerphi[/URL](b)) values of bases b
1 2 ("base 1" does not exist) [solved] [number of primes: 1] [length of largest prime: 2]
2 3, 4, 6 [all solved] [number of primes: 3, 5, 11] [length of largest prime: 3, 3, 5]
4 5, 8, 10, 12 [all solved] [number of primes: 22, 75, 77, 106] [length of largest prime: 96, 221, 31, 42]
6 7, 9, 14, 18 [all solved] [number of primes: 71, 151, 650, 549] [length of largest prime: 17, 1161, 19699, 6271]
8 15, 16, 20, 24, 30 [all are solved except 16] [number of primes: 1284, 2345~2347, 3314, 3409, 2619] [length of largest prime: 157, 32235 or >50000, 6271, 8134, 34206]
10 11, 22 [all solved] [number of primes: 1068, 8003] [length of largest prime: 62669, 22003]
12 13, 21, 26, 28, 36, 42
16 17, 32, 34, 40, 48, 60
18 19, 27, 38, 54
20 25, 33, 44, 50, 66
22 23, 46
24 35, 39, 45, 52, 56, 70, 72, 78, 84, 90
28 29, 58
30 31, 62
32 51, 64, 68, 80, 96, 102, 120
36 37, 57, 63, 74, 76, 108, 114, 126
40 41, 55, 75, 82, 88, 100, 110, 132, 150
42 43, 49, 86, 98
44 69, 92, 138
46 47, 94
48 65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210
52 53, 106
54 81, 162
56 87, 116, 174
58 59, 118
60 61, 77, 93, 99, 122, 124, 154, 186, 198
64 85, 128, 136, 160, 170, 192, 204, 240
[/CODE]

(see the text file for more values, up to level 2560)

Condensed table for bases 2 <= b <= 16 and b = 18, 20, 22, 24, 30 (bases 11, 13, 16, 22, 30 data assume the primality of the strong probable primes)

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×b^n+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
9 151 3(0^1158)11 1161 3×9^1160+10
10 77 5(0^28)27 31 5×10^30+27
11 1068 5(7^62668) 62669 (57×11^62668−7)/10
12 106 4(0^39)77 42 4×12^41+91
13 3195~3197 8(0^32017)111 32021 8×13^32020+183
14 650 4(D^19698) 19699 5×14^19698−1
15 1284 (7^155)97 157 (15^157+59)/2
16 2346~2347 (4^72785)DD 72787 (4×16^72787+2291)/15
18 549 C(0^6268)5C 6271 12×18^6270+221
20 3314 G(0^6269)D 6271 16×20^6270+13
22 8003 B(K^22001)5 22003 (251×22^22002−335)/21
24 3409 N00(N^8129)LN 8134 13249×24^8131−49
30 2619 O(T^34205) 34206 25×30^34205−1
[/CODE]

sweety439 2021-06-03 08:59

New minimal prime (start with b+1) in base b is found for b=650: 3:{649}^(498101), see [URL="https://mersenneforum.org/showpost.php?p=579849&postcount=931"]https://mersenneforum.org/showpost.php?p=579849&postcount=931[/URL]

Added it to excel file [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]

Base 108 is an interesting base since ....

* For the family {1}, length 2 is prime, but the next prime is large (length 449)
* For the family 1{0}1, length 2 is prime, the next prime is not known
* For the family y{z}, first prime is large (length 411)
* For the family 11{0}1, first prime is large (length 400)
* For the family {y}z, first prime is large (length 492) (note that length 1 is also prime, but length 1 is not allowed in this project)
* For the family 6{0}1, first prime is large (length 16318)
* For the family #{z} (# = (base/2)-1)), first prime is large (length 7638)

This situation is not common in bases with many divisors, but although 108 has many divisors, this situation occurs in this base, this is why this base is interesting :))

Also base 282 ....

* For the family A{0}1, first prime is large (length 1474)
* For the family C{0}1, first prime is large (length 2957)
* For the family z{0}1, first prime is large (length 277)

[URL="http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base282-reserve.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base282-reserve.htm[/URL] only tells you that all these three families have a prime with length <= 100001 ....

* For the family 7{z}, first prime is large (length 21413)
* For the family 10{z}, first prime is large (length 780)

[URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base282-reserve.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base282-reserve.htm[/URL] only tells you that the farmer family has a prime with length <= 100001, and the letter family has a prime with length <= 100002

Some extremely low weight bases are 383, 458, 578, 647, 698, 773, 938, 992

* In base 383, families 2{0}1 and its dual family 1{0}2 have very low weight (smaller than the weight of the Fermat number (i.e. 1{0}1 in base 2))
* In base 458, families y{z}, A{0}1, 9{z}, A{z} do not have easy primes
*

sweety439 2021-06-20 04:21

Searched 1{0}2 (b^n+2) and {z}y (b^n-2) (for bases 2<=b<=1024) up to n=5000 and found the (probable) primes 485^3164-2 and 487^3775-2

b^n+2 for all remain bases b<=711 and b^n-2 for all remain bases b<=533 are checked to n=5000 with no (probable) primes found.

I will reserve 1{0}z (b^n+(b-1)) and {z}1 (b^n-(b-1)) (for bases 2<=b<=1024) (also up to n=5000) after this reservation was done.

These families were already tested to large n: (only consider families which [B]must[/B] be minimal primes (start with b+1))

{1}: [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html[/URL] [URL="https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt"]https://raw.githubusercontent.com/xayahrainie4793/Sierpinski-Riesel-for-fixed-k-and-variable-base/master/Riesel%20k1.txt[/URL]

1{0}1: [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL] [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL]

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1: [URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL]

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}: [URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL]

z{0}1: [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL]

y{z}: [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL] [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL]

1{0}z: [URL="https://oeis.org/A076845"]https://oeis.org/A076845[/URL]

{z}1: [URL="https://oeis.org/A113516"]https://oeis.org/A113516[/URL]

1{0}2: [URL="http://oeis.org/A138066"]http://oeis.org/A138066[/URL]

{z}y: [URL="https://www.primepuzzles.net/puzzles/puzz_887.htm"]https://www.primepuzzles.net/puzzles/puzz_887.htm[/URL] [URL="http://oeis.org/A255707"]http://oeis.org/A255707[/URL]

and [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]this table[/URL] was updated

Formula of these families:

1{0}1: b^n+1 (b>=2) (n>=1) (length=n+1)
1{0}2: b^n+2 (b>=3) (n>=1) (length=n+1)
1{0}3: b^n+3 (b>=4) (n>=1) (length=n+1)
1{0}4: b^n+4 (b>=5) (n>=1) (length=n+1)
1{0}z: b^n+(b-1) (b>=2) (n>=1) (length=n+1)
{1}: (b^n-1)/(b-1) (b>=2) (n>=2) (length=n)
1{2}: ((b+1)*b^n-2)/(b-1) (b>=3) (n>=1) (length=n+1)
1{3}: ((b+2)*b^n-3)/(b-1) (b>=4) (n>=1) (length=n+1)
1{4}: ((b+3)*b^n-4)/(b-1) (b>=5) (n>=1) (length=n+1)
1{z}: 2*b^n-1 (b>=2) (n>=1) (length=n+1)
2{0}1: 2*b^n+1 (b>=3) (n>=1) (length=n+1)
2{0}3: 2*b^n+3 (b>=4) (n>=1) (length=n+1)
{2}1: (2*b^n-(b+1))/(b-1) (b>=3) (n>=2) (length=n)
2{z}: 3*b^n-1 (b>=3) (n>=1) (length=n+1)
3{0}1: 3*b^n+1 (b>=4) (n>=1) (length=n+1)
3{0}2: 3*b^n+2 (b>=4) (n>=1) (length=n+1)
3{0}4: 3*b^n+4 (b>=5) (n>=1) (length=n+1)
{3}1: (3*b^n-(2*b+1))/(b-1) (b>=4) (n>=2) (length=n)
3{z}: 4*b^n-1 (b>=4) (n>=1) (length=n+1)
4{0}1: 4*b^n+1 (b>=5) (n>=1) (length=n+1)
4{0}3: 4*b^n+3 (b>=5) (n>=1) (length=n+1)
{4}1: (4*b^n-(3*b+1))/(b-1) (b>=5) (n>=2) (length=n)
4{z}: 5*b^n-1 (b>=5) (n>=1) (length=n+1)
5{0}1: 5*b^n+1 (b>=6) (n>=1) (length=n+1)
5{z}: 6*b^n-1 (b>=6) (n>=1) (length=n+1)
6{0}1: 6*b^n+1 (b>=7) (n>=1) (length=n+1)
6{z}: 7*b^n-1 (b>=7) (n>=1) (length=n+1)
7{0}1: 7*b^n+1 (b>=8) (n>=1) (length=n+1)
7{z}: 8*b^n-1 (b>=8) (n>=1) (length=n+1)
8{0}1: 8*b^n+1 (b>=9) (n>=1) (length=n+1)
8{z}: 9*b^n-1 (b>=9) (n>=1) (length=n+1)
9{0}1: 9*b^n+1 (b>=10) (n>=1) (length=n+1)
9{z}: 10*b^n-1 (b>=10) (n>=1) (length=n+1)
A{0}1: 10*b^n+1 (b>=11) (n>=1) (length=n+1)
A{z}: 11*b^n-1 (b>=11) (n>=1) (length=n+1)
B{0}1: 11*b^n+1 (b>=12) (n>=1) (length=n+1)
B{z}: 12*b^n-1 (b>=12) (n>=1) (length=n+1)
C{0}1: 12*b^n+1 (b>=13) (n>=1) (length=n+1)
{#}$ (# = (b−1)/2, $ = (b+1)/2): (b^n+1)/2 (b>=3 is odd) (n>=2) (length=n)
{y}z: ((b-2)*b^n+1)/(b-1) (b>=3) (n>=2) (length=n)
y{z}: (b-1)*b^n-1 (b>=3) (n>=1) (length=n+1)
z{0}1: (b-1)*b^n+1 (b>=2) (n>=1) (length=n+1)
{z}1: b^n-(b-1) (b>=2) (n>=2) (length=n)
{z}w: b^n-4 (b>=5) (n>=2) (length=n)
{z}x: b^n-3 (b>=4) (n>=2) (length=n)
{z}y: b^n-2 (b>=3) (n>=2) (length=n)

sweety439 2021-06-21 00:15

Records of length in these families for various bases: (format: base (length)) (bases with "NB" or "RC" for given family are not counted)

1{0}1:

2 (2)
14 (3)
34 (5)
38 (>8388608)

1{0}2:

3 (2)
23 (12)
47 (114)
89 (256)
167 (>100001)

1{0}3:

4 (2)
22 (3)
32 (4)
46 (21)
292 (40)
382 (256)
530 (1399)
646 (>5000)

1{0}4:

5 (3)
23 (7)
53 (13403)
139 (>25000)

1{0}z:

2 (2)
5 (3)
14 (17)
32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
173? (>5000)

1{0}11:

2 (3)
9 (4)
11 (5)
23 (10)
35 (16)
63 (74)
68 (596)
198 (5198)
213? (>5000)

10{z}:

2 (3)
7 (6)
23 (9)
42 (10)
63 (1485)
88 (1706)
208 (26682)
575 (>247001)

11{0}1:

2 (3)
9 (4)
18 (11)
51 (185)
63 (187)
108 (400)
171 (1853)
201 (31276)
222 (52727)
327 (135983)
813? (>100000)

{1}0z:

2 (3)
7 (61)
27 (97)
47 (565)
57 (1109)
137 (3953)
161 (9155)
167? (>5000)

{1}:

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)
51 (4229)
91 (4421)
152 (270217)
185? (>66337)

{1}2:

3 (2)
7 (4)
19 (42)
25 (118)
31 (2056)
61 (2128)
91 (3096)
93 (>5000)

1{2}:

3 (2)
7 (4)
31 (76)
97 (1128)
265 (2301)
355 (>5000)

1{3}:

4 (2)
5 (3)
17 (5)
29 (19)
46 (82)
59 (85)
71 (197)
107 (>5000)

1{4}:

5 (5)
11 (19)
17 (61)
83 (>5000)

1{z}:

2 (2)
5 (5)
20 (11)
29 (137)
67 (769)
107 (21911)
170 (166429)
581 (>400001)

2{0}1:

3 (2)
12 (4)
17 (48)
38 (2730)
101 (192276)
218 (333926)
365? (>300001)

2{0}3:

4 (2)
23 (3)
44 (4)
58 (6)
59 (18)
79 (>5000)

2{1}:

3 (2)
4 (3)
7 (4)
13 (16)
19 (24)
37 (36)
47 (76)
77 (476)
85 (6940)
117? (>5000)

{2}1:

3 (2)
4 (3)
10 (4)
28 (40)
31 (1987)
106 (>5000)

2{z}:

4 (2)
12 (3)
32 (12)
42 (2524)
432 (16003)
588 (>500001)

3{0}1:

4 (2)
8 (3)
18 (4)
28 (8)
44 (10)
62 (13)
72 (15)
108 (271)
314 (281)
358 (9561)
718 (>300001)

3{0}2:

5 (2)
25 (3)
39 (24)
47 (28)
99 (104)
109 (958)
223 (>5000)

3{0}4:

5 (2)
7 (3)
17 (11)
61 (29)
97 (1924)
167 (>5000)

{3}1:

4 (2)
5 (5)
7 (17)
16 (25)
19 (221)
31 (4261)
79 (>5000)

3{z}:

5 (2)
23 (6)
47 (1556)
72 (1119850)
275? (>600001)

4{0}1:

5 (3)
17 (7)
23 (343)
32 (>1717986918)

4{0}3:

5 (2)
13 (204)
83 (>5000)

{4}1:

5 (5)
8 (9)
11 (45)
46 (>10000)

4{z}:

6 (2)
8 (5)
14 (19699)
338 (>300001)

5{0}1:

6 (2)
18 (3)
24 (13)
44 (16)
60 (43)
122 (136)
170 (176)
200 (768)
308 (>300001)

5{z}:

7 (2)
13 (3)
37 (4)
48 (295)
119 (666)
154 (1990)
234 (>600001)

6{0}1:

7 (2)
9 (3)
14 (7)
19 (15)
20 (16)
48 (28)
53 (144)
67 (4533)
108 (16318)
129 (16797)
212 (>500001)

6{z}:

8 (4)
38 (8)
68 (25396)
308 (>300001)

7{0}1:

8 (3)
24 (4)
32 (5)
50 (517)
224 (689)
338 (793)
398 (17473)
1004 (54849)
1136? (beyond the base limit (b=1024), thus not searched)

7{z}:

9 (2)
11 (3)
35 (5)
42 (11)
44 (17)
47 (33)
68 (63)
97 (192336)
321 (>500001)

8{0}1:

9 (2)
15 (3)
23 (119216)
53 (227184)
86 (>1000001)

8{z}:

10 (2)
18 (12)
38 (44)
88 (172)
112 (5718)
138 (35686)
378 (>300001)

9{0}1:

10 (4)
24 (6)
32 (14)
38 (22)
74 (66)
94 (264)
244 (1836)
248 (39511)
592 (96870)
724 (>400001)

9{z}:

11 (2)
12 (3)
17 (118)
80 (>400001)

A{0}1:

11 (11)
17 (1357)
101 (1507)
173 (264235)
185 (>1000001)

A{z}:

12 (2)
20 (9)
30 (31)
38 (767)
72 (2446)
214 (>1000001)

B{0}1:

12 (4)
48 (8)
50 (10)
68 (3948)
542 (4910)
560 (>100001)

B{z}:

13 (3)
18 (9)
31 (73)
43 (204)
65 (1194)
98 (3600)
153 (21660)
186 (112718)
263 (>314001)

C{0}1:

13 (2)
17 (3)
21 (11)
24 (43)
30 (1024)
68 (656922)
163? (>500001)

{y}z:

3 (2)
8 (3)
13 (564)
83 (680)
143 (>5000)

y{z}:

3 (2)
8 (4)
15 (15)
23 (56)
26 (134)
38 (136212)
113 (286644)
128 (>2450001)

z{0}1:

2 (2)
5 (3)
10 (4)
11 (11)
19 (30)
41 (81)
53 (961)
88 (3023)
122 (6217)
123 (>400001)

{z0}z1:

2 (2)
5 (4)
14 (6)
19 (16)
30 (138)
50 (1152)
53 (21942)
97 (>500000)

{z}yz:

2 (3)
13 (4)
19 (5)
33 (7)
37 (9)
43 (31)
52 (108)
99 (131)
190 (562)
213 (643)
215 (22342)
517? (>5000)

{z}1:

2 (2)
5 (5)
8 (13)
20 (17)
29 (33)
37 (67)
71 (3019)
93 (>60000)

{z}w:

5 (5)
27 (7)
35 (13)
47 (65)
65 (175)
123 (299)
141 (395)
207 (>5000)

{z}x:

4 (2)
16 (3)
22 (6)
28 (10)
50 (21)
52 (105)
94 (204)
152 (346)
154 (396)
302 (1061)
478 (1410)
512 (1600)
542 (1944)
1192? (beyond the base limit (b=1024), thus not searched)

{z}y:

3 (2)
11 (4)
17 (6)
23 (24)
79 (38)
81 (130)
97 (747)
287 (3410)
305 (>30000)

sweety439 2021-06-23 04:57

4 Attachment(s)
all b^n+-2 and b^n+-(b-1) for 2<=b<=1024 tested to n=5000

status files attached

(z means b-1, y means b-2)

b^n+2 = (1000...0002) base b = 10002
b^n-2 = (zzz...zzzy) base b = zzzzy
b^n+(b-1) = (1000...000z) base b = 1000z
b^n-(b-1) = (zzz...zzz1) base b = zzzz1

edit: ((b-2)*b^n+1)/(b-1) (family yyyyz) also tested to n=5000

sweety439 2021-06-24 01:52

There are many conjectures related to this project (find all minimal primes (start with b+1) in bases 2<=b<=1024):

* Are there infinitely many [URL="https://primes.utm.edu/glossary/xpage/Mersennes.html"]Mersenne primes[/URL]? (related to family {1} in base 2)
* Are there infinitely many [URL="https://primes.utm.edu/glossary/xpage/FermatNumber.html"]Fermat primes[/URL]? (related to family 1{0}1 in base 2)
* Are there infinitely many Wagstaff primes? (related to family {2}3 in base 4)
* Are there infinitely many repunit primes? (related to family {1} in base 10)
* Are there infinitely many generalized Fermat primes base 10? (related to family 1{0}1 in base 10)
* [URL="https://web.archive.org/web/20120426061657/http://oddperfect.org/"]Odd perfect numbers search[/URL] (related to family {1} in prime bases)
* [URL="http://www.acta.sapientia.ro/acta-math/C1-1/MATH1-6.PDF"]n-hyperperfect numbers search[/URL] (related to family {z}1 in base n+1 if n+1 is prime)
* [URL="https://oeis.org/A305237"]Are there infinitely many triples of 3 consecutive numbers with all have primitive roots?[/URL] (related to families {1}, {2}1, {1}2, 1{0}2, 1{2}, 2{0}1 in base 3)
* [URL="https://primes.utm.edu/glossary/xpage/NewMersenneConjecture.html"]New Mersenne conjecture[/URL] (primes p in families {1} in base 2, 1{0}1 in base 2, {3}1 in base 4, 1{0}3 in base 4, and related to Mersenne primes ({1} in base 2) and Wagstaff primes ({2}3 in base 4))
* "Dividing Phi" category (related to family 2{0}1 in bases == 11 mod 12)
* Sierpinski problem (related to family *{0}1 in base 2)
* Riesel problem (related to family *{1} in base 2)
* Dual Sierpinski problem (related to family 1{0}* in base 2)
* Dual Riesel problem (related to family {1}* in base 2)
* Generalized Sierpinski problem base b (related to family *{0}1 in base b)
* Generalized Riesel problem base b (related to family *{z} in base b)
* [URL="http://www.worldofnumbers.com/em197.htm"]Problem 197[/URL] (related to family *{1} in base 10)

Also related types of primes (left-truncatable primes, right-truncatable primes, two-sides primes, detelable primes, permutable primes, circular primes, palindromic primes, etc.)

This project specially to [I]prime[/I] bases are more related to the conjectures because.... (since prime bases are more important to number theory (like that Mersenne primes are more important to number theory than palindromic primes, since the former is related to perfect numbers), if "prime = base (b)" (i.e. the prime "10" in this base) is also included will make prime bases more uninteresting then other bases since the "10" is already prime and all larger numbers with "10" as subsequence will be excluded, and this is bad, thus my project use [URL="https://mersenneforum.org/showpost.php?p=531632&postcount=7"]LaurV's suggestion[/URL], i.e. the "prime = base (b)" (i.e. the prime "10" in this base) is excluded as the primes < base (b) (i.e. the single-digit primes, which are really trivial), since including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, also, if we include the prime = b (i.e. the prime "10") when the base (b) is prime, then some properties in [URL="https://mersenneforum.org/showpost.php?p=593116&postcount=208"]this post[/URL] will be incorrect, also, start with b+1 (instead of b) makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base b more simple and [URL="https://en.wikipedia.org/wiki/Smooth_number"]smooth number[/URL], for more reasons, see [URL="https://mersenneforum.org/showpost.php?p=595887&postcount=250"]this post[/URL]

* {1} family, related to [URL="http://oddperfect.org/"]odd perfect numbers search[/URL] (broken link: [URL="https://web.archive.org/web/20120426061657/http://oddperfect.org/"]from wayback machine[/URL]) (factorization of numbers in this family) (in fact, factorization of numbers in this family is related to [URL="https://en.wikipedia.org/wiki/Divisor_function"]sum-of-divisors function[/URL], thus not only related to odd perfect numbers search, but also related to [URL="https://en.wikipedia.org/wiki/Amicable_numbers"]amicable numbers[/URL] search, [URL="https://en.wikipedia.org/wiki/Quasiperfect_number"]quasiperfect numbers[/URL] search, [URL="https://en.wikipedia.org/wiki/Betrothed_numbers"]betrothed numbers[/URL] search, and [URL="https://en.wikipedia.org/wiki/Aliquot_sequence"]Aliquot sequence[/URL]), also, there is [URL="http://myfactorcollection.mooo.com:8090/oddperfect/Mar12_2021/opfactors.gz"]opfactors.gz[/URL] only for prime bases in page [URL="http://myfactorcollection.mooo.com:8090/downloads.html"]http://myfactorcollection.mooo.com:8090/downloads.html[/URL], also researched in [URL="https://oeis.org/A065854"]https://oeis.org/A065854[/URL] and [URL="https://oeis.org/A279068"]https://oeis.org/A279068[/URL]
* 2{0}1 family, related to "[URL="https://mersenneforum.org/showthread.php?t=19725"]Dividing Phi" category[/URL]" (when the prime is == 11 mod 12) (and when the prime is == 5 mod 12, it will divide Phi(2*p^n,2) instead of Phi(p^n,2))
* z{0}1 family, related to inverse of [URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]Euler totient function[/URL] ([URL="https://oeis.org/A087126"]it is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime[/URL]) (the same holds for y{z} family related to inverse of [URL="http://en.wikipedia.org/wiki/Dedekind_psi_function"]Dedekind psi function[/URL], i.e. it is usually the case that, for prime p and k > 1, the first time the Dedekind psi function psi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) - 1 is prime, references: [URL="https://oeis.org/A000010"]https://oeis.org/A000010[/URL] (Euler totient function), [URL="https://oeis.org/A001615"]https://oeis.org/A001615[/URL] (Dedekind psi function), [URL="https://oeis.org/A002202"]https://oeis.org/A002202[/URL] (range of Euler totient function) [URL="https://oeis.org/A203444"]https://oeis.org/A203444[/URL] (range of Dedekind psi function), [URL="https://oeis.org/A087139"]https://oeis.org/A087139[/URL], [URL="https://oeis.org/A122396"]https://oeis.org/A122396[/URL])
* y{z} family, researched in [URL="http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf"]http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf[/URL] [URL="https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf"]https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf[/URL]
* {z}1 family, related to [URL="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html"](this prime minus 1)-hyperperfect numbers search[/URL]
* {#}$ family (# = (b−1)/2, $ = (b+1)/2), researched in [URL="https://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2009-12-013_R1.pdf"]https://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2009-12-013_R1.pdf[/URL] and [URL="https://oeis.org/A341211"]https://oeis.org/A341211[/URL]
* *{0}1 family and *{z} family (* is any string of digits), researched in [URL="https://www.jstor.org/stable/2005886"]https://www.jstor.org/stable/2005886[/URL]

also, for the large primes in *{0}1 family (which can be proven prime using [URL="https://primes.utm.edu/prove/prove3_1.html"]N-1 method[/URL]) and *{z} family (which can be proven prime using [URL="https://primes.utm.edu/prove/prove3_2.html"]N+1 method[/URL]), the case which the base (b) is prime are proved most quickly, and if [URL="https://oeis.org/A001222"]bigomega[/URL](b) is larger, then the prove is slower, since in this case N-1 or N+1 has more prime factors (this is why the prove of n! +/- 1 primes and of p# +/- 1 primes are very slow).

also, the prime primitive root mod p: [URL="https://oeis.org/A002233"]https://oeis.org/A002233[/URL] [URL="https://oeis.org/A103309"]https://oeis.org/A103309[/URL], is more important in number theory because [URL="https://oeis.org/A223942"]https://oeis.org/A223942[/URL]

also, the ring of the [URL="https://en.wikipedia.org/wiki/P-adic_number"]b-adic numbers[/URL] (related to base b numbers, see [URL="https://en.wikipedia.org/wiki/Automorphic_number"]https://en.wikipedia.org/wiki/Automorphic_number[/URL]) is a field if and only if b is prime or prime power (thus the b-adic numbers generally used in mathematics only for prime b), also, the multiplicative order of the base (b) mod primes (i.e. znorder(Mod(b,p)) with prime p) is important in this problem (see post [URL="https://mersenneforum.org/showpost.php?p=582061&postcount=154"]#154[/URL]), and the multiplicative order of the base (b) mod primes (i.e. znorder(Mod(b,p)) with prime p) is more important in number theory when b is prime (see [URL="https://oeis.org/A212953"]https://oeis.org/A212953[/URL], [URL="https://oeis.org/A218356"]https://oeis.org/A218356[/URL], [URL="https://oeis.org/A218357"]https://oeis.org/A218357[/URL], [URL="https://oeis.org/A218358"]https://oeis.org/A218358[/URL], [URL="https://oeis.org/A218359"]https://oeis.org/A218359[/URL], [URL="https://oeis.org/A213224"]https://oeis.org/A213224[/URL], they are analogs of [URL="https://oeis.org/A003060"]https://oeis.org/A003060[/URL] in prime bases)

also, the generalized Wieferich primes only for prime bases b (see post [URL="https://mersenneforum.org/showpost.php?p=582061&postcount=154"]#154[/URL]) are also more important since they are related to [URL="https://en.wikipedia.org/wiki/Wieferich_pair"]Wieferich pair[/URL] and [URL="http://wayback.cecm.sfu.ca/~mjm/WieferichBarker/Data/AllCycles.txt"]Barker sequence[/URL], and hence related to [URL="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem"]Fermat's Last Theorem[/URL] and [URL="https://en.wikipedia.org/wiki/Catalan%27s_conjecture"]Catalan's conjecture[/URL], there are pages for the generalized Wieferich primes only for prime bases b: [URL="http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html"]http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html[/URL] (broken link: [URL="https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vSE8z4_MlxiVUSvsNMfCMo_mCwQqxqJAIzfPH3JjFP2c6T-VtRAGt8dyG8rTUnK9L_DTJy3cPMO8B3p/pub"]cached copy[/URL]) [URL="http://home.earthlink.net/~oddperfect/FermatQuotients.html"]http://home.earthlink.net/~oddperfect/FermatQuotients.html[/URL] (broken link: [URL="https://web.archive.org/web/20160417130531/http://home.earthlink.net/~oddperfect/FermatQuotients.html"]from wayback machine[/URL] [URL="https://docs.google.com/document/d/e/2PACX-1vQFlX9GM3r5v3ocFoE1AkgLlXtzbyg8Ga9-AbNgNlgFuJ_Ti-dZB7XB9qLKj6k5rmCRB2i1I2jSlUhp/pub"]cached copy[/URL])

sweety439 2021-06-27 13:19

We can use the sense of [URL="http://www.iakovlev.org/zip/riesel2.pdf"]http://www.iakovlev.org/zip/riesel2.pdf[/URL], [URL="https://scholar.google.com/scholar_url?url=https://www.sciencedirect.com/science/article/pii/S0022314X08000462/pdf%3Fmd5%3Dcb11465f6eb6873d749c67b2b31dbb1d%26pid%3D1-s2.0-S0022314X08000462-main.pdf%26_valck%3D1&hl=zh-TW&sa=T&oi=ucasa&ct=ufr&ei=TnR0YYi5IoP2yASq-aXQCA&scisig=AAGBfm1x9DSu578ydrXxfMnrRUPp1l8rcA"]https://scholar.google.com/scholar_url?url=https://www.sciencedirect.com/science/article/pii/S0022314X08000462/pdf%3Fmd5%3Dcb11465f6eb6873d749c67b2b31dbb1d%26pid%3D1-s2.0-S0022314X08000462-main.pdf%26_valck%3D1&hl=zh-TW&sa=T&oi=ucasa&ct=ufr&ei=TnR0YYi5IoP2yASq-aXQCA&scisig=AAGBfm1x9DSu578ydrXxfMnrRUPp1l8rcA[/URL], [URL="https://people.math.sc.edu/filaseta/papers/SierpinskiEtCoPapNew.pdf"]https://people.math.sc.edu/filaseta/papers/SierpinskiEtCoPapNew.pdf[/URL], [URL="https://mersenneforum.org/showpost.php?p=285767&postcount=65"]https://mersenneforum.org/showpost.php?p=285767&postcount=65[/URL], [URL="https://mersenneforum.org/showpost.php?p=138737&postcount=24"]https://mersenneforum.org/showpost.php?p=138737&postcount=24[/URL], [URL="https://mersenneforum.org/showpost.php?p=153508&postcount=147"]https://mersenneforum.org/showpost.php?p=153508&postcount=147[/URL], [URL="https://mersenneforum.org/showpost.php?p=155243&postcount=176"]https://mersenneforum.org/showpost.php?p=155243&postcount=176[/URL], [URL="https://mersenneforum.org/showthread.php?t=11143"]https://mersenneforum.org/showthread.php?t=11143[/URL], [URL="https://mersenneforum.org/showpost.php?p=549958&postcount=867"]https://mersenneforum.org/showpost.php?p=549958&postcount=867[/URL], [URL="https://mersenneforum.org/showpost.php?p=550208&postcount=883"]https://mersenneforum.org/showpost.php?p=550208&postcount=883[/URL], [URL="https://mersenneforum.org/showpost.php?p=550364&postcount=891"]https://mersenneforum.org/showpost.php?p=550364&postcount=891[/URL], [URL="https://mersenneforum.org/showpost.php?p=550372&postcount=893"]https://mersenneforum.org/showpost.php?p=550372&postcount=893[/URL], [URL="https://stdkmd.net/nrr/1/11113.htm#prime_period"]https://stdkmd.net/nrr/1/11113.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/13333.htm#prime_period"]https://stdkmd.net/nrr/1/13333.htm#prime_period[/URL], [URL="https://stdkmd.net/nrr/1/10003.htm#prime_period"]https://stdkmd.net/nrr/1/10003.htm#prime_period[/URL], [URL="https://mersenneforum.org/showpost.php?p=452132&postcount=66"]https://mersenneforum.org/showpost.php?p=452132&postcount=66[/URL] ("Mersenne number" can be generated to generalized repunit number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(y*n+z)-1)/(x-1) where x is a root of the base (b)) with x>2 (the "Mersenne number" is the x=2 case) and generalized Wagstaff number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(2*(y*n+z)+1)+1)/(x+1) where x is a root of the base (b)), and "GFN" can be generated to generalized half Fermat number (i.e. (a*b^n+c)/gcd(a+c,b-1) can be written as (x^(y*n+z)+1)/2 where x is a root of the base (b)) with odd x (the "GFN" is x^(y*n+z)+1 with even x)), to conclude that the unsolved families (unsolved families are families which are neither primes (>base) found nor can be proven to contain no primes > base) eventually should yield a prime, this can be calculated for the [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] (or the [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]), families which can be proven to contain no primes > base have Nash weight (or difficulty) 0, e.g. for the base 11 unsolved family 5{7}:

5(7^n) = (57*11^n-7)/10, but there is no n satisfying that 57*11^n and 7 are both r-th powers for some r>1 (since 7 is not perfect power), nor there is n satisfying that 57*11^n and -7 are (one is 4th power, another is of the form 4*m^4) (since -7 is neither 4th power nor of the form 4*m^4), thus, 5(7^n) has no algebra factors for any n, thus 5(7^n) eventually should yield a prime unless it can be proven to contain no primes > base using covering congruence, and we have:

5(7^n) is divisible by 2 for n == 1 mod 2
5(7^n) is divisible by 13 for n == 2 mod 12
5(7^n) is divisible by 17 for n == 4 mod 16
5(7^n) is divisible by 5 for n == 0 mod 5
5(7^n) is divisible by 23 for n == 6 mod 22
5(7^n) is divisible by 601 for n == 8 mod 600
5(7^n) is divisible by 97 for n == 12 mod 48
5(7^n) is divisible by 1279 for n == 16 mod 426
...

and it does not appear to be any covering set of primes (and its Nash weight (or difficulty) is positive, and it has prime candidate), so there must be a prime at some point.

(see post [URL="https://mersenneforum.org/showpost.php?p=568675&postcount=103"]#103[/URL] for examples of families which can be proven to contain no primes > base)

The multiplicative order of the base (b) mod primes (i.e. znorder(Mod(b,p)) with prime p) is important in this problem, since all primes in the [URL="http://irvinemclean.com/maths/siercvr.htm"]covering set[/URL] with period n in base b are prime factors of the [URL="https://primes.utm.edu/glossary/xpage/GeneralizedRepunitPrime.html"]generalized repunit[/URL] (b^n-1)/(b-1), and a prime p is prime factor of the generalized repunit (b^n-1)/(b-1) if and only if the multiplicative order of b mod p divides n and > 1, or p divides b-1 (hence the multiplicative order of b mod p is 1) and p divides n, an important theorem is if a == b ([URL="https://en.wikipedia.org/wiki/Modulo_(mathematics)"]mod[/URL] p), then multiplicative order of a mod p is equal to multiplicative order of b mod p (this is even true if p is not prime, the only condition is gcd(a,p) = gcd(b,p) = 1), since if a prime p divides the number with n digits in a family in base b, then p also divides the number with k*r+n digits in the same family in base b for all nonnegative integer k, where r is the multiplicative order of b mod p (unless the multiplicative order of b mod p is 1, i.e. p divides b-1, in this case p also divides the number with k*p+n digits in the same family in base b for all nonnegative integer k), also, the multiplicative order of b mod p must divide p-1 (by [URL="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem"]Fermat's little theorem[/URL]), and the number of 0<=b<=p-1 (i.e. the number of b in Z_p) such that the multiplicative order of b mod p is r is ([URL="https://oeis.org/A000010"]eulerphi[/URL](r) if r divides p-1, 0 if r does not divide p-1) (also note that for two primes p and q (this is even true if p or/and q are not primes, the only condition is gcd(p,q) = 1), b mod p and b mod q are completely [URL="https://en.wikipedia.org/wiki/Independent_variables"]independent variables[/URL]), and if the multiplicative order of b mod p is exactly p-1 (i.e. b is [URL="https://en.wikipedia.org/wiki/Primitive_root_modulo_n"]primitive root[/URL] mod p), then p is [URL="https://en.wikipedia.org/wiki/Full_reptend_prime"]full reptend prime[/URL] base b (there are [URL="https://oeis.org/A000010"]eulerphi[/URL](p-1) primitive roots b with 0<=b<=p-1 (i.e. b in Z_p), there is a [URL="http://www.bluetulip.org/2014/programs/primitive.html"]website[/URL] to calculate all primitive roots of p), and thus this problem is related to [URL="https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots"]Artin's conjecture on primitive roots[/URL] (for the research of the smallest primitive root of primes, see [URL="https://oeis.org/A023048"]https://oeis.org/A023048[/URL] and [URL="http://sweet.ua.pt/tos/p_roots.html"]http://sweet.ua.pt/tos/p_roots.html[/URL], and there is a conjecture that all primes p > 409 have a primitive root > sqrt(p), see [URL="https://oeis.org/A262264"]https://oeis.org/A262264[/URL], also see the thread [URL="https://mersenneforum.org/showthread.php?t=18797"]https://mersenneforum.org/showthread.php?t=18797[/URL]), also, the primes p such that the multiplicative order of b mod p is n (such primes p are always == 1 mod n, for references for b=2, see: [URL="https://www.mersenne.org/various/math.php#trial_factoring"]n is prime[/URL] [URL="http://www.doublemersennes.org/math.php"]n is Mersenne prime[/URL] [URL="http://www.prothsearch.com/fermat.html"]n is power of 2[/URL] [URL="https://www.alpertron.com.ar/MODFERM.HTM"]n is power of 3 or twice power of 3[/URL] [URL="https://mersenneforum.org/showpost.php?p=383931&postcount=1"]n is prime power[/URL]) are exactly the primes p dividing Zs(n,b,1), where Zs is the [URL="https://en.wikipedia.org/wiki/Zsigmondy's%20theorem"]Zsigmondy number[/URL], i.e. Zs(n,b,1) is the greatest divisor of b^n - 1 that is coprime to b^m - 1 for all positive integers m < n, with b>=2 and n>=1 (the Zsigmondy number Zs(n,b,1) is equal to Phi_n(b)/gcd(Phi_n(b),n) if n != 2, where Phi is [URL="https://en.wikipedia.org/wiki/Cyclotomic_polynomial"]cyclotomic polynomial[/URL], if n = 2, then Zs(n,b,1) = [URL="https://oeis.org/A000265"]A000265[/URL](b+1), for the table of Zs(n,b,1), see [URL="https://oeis.org/A323748"]https://oeis.org/A323748[/URL]), if (and only if) there is only one such prime, then this prime is [URL="https://en.wikipedia.org/wiki/Unique_prime"]unique prime[/URL] in base b ([URL="https://primes.utm.edu/top20/page.php?id=44"]generalized unique prime[/URL] base b), also, the [URL="https://en.wikipedia.org/wiki/Aurifeuillean_factorization"]Aurifeuillean factors[/URL] of b^n+-1 (if n is an odd multiple of [URL="https://oeis.org/A007913"]A007913[/URL](b), i.e. n/[URL="https://oeis.org/A007913"]A007913[/URL](b) is an odd integer, then b^n-1 has Aurifeuillean factorization if [URL="https://oeis.org/A007913"]A007913[/URL](b) == 1 mod 4, b^n+1 has Aurifeuillean factorization if [URL="https://oeis.org/A007913"]A007913[/URL](b) == 2, 3 mod 4, this is the reason why [URL="http://www.numericana.com/answer/constants.htm#artin"]the density of the primes p such that b is primitive root mod p is not just C_Artin (but a rational multiple of C_Artin) if A007913(b) == 1 mod 4[/URL]), the coefficient of the Aurifeuillean factors are the [URL="http://myfactorcollection.mooo.com:8090/LCD_2_998"]Lucas C,D polynomials[/URL] of A007913(b), and for b=2 this is [URL="https://en.wikipedia.org/wiki/Gaussian_prime"]Gaussian prime[/URL] (see [URL="https://oeis.org/A124112"]https://oeis.org/A124112[/URL], [URL="https://oeis.org/A125742"]https://oeis.org/A125742[/URL], [URL="https://oeis.org/A124165"]https://oeis.org/A124165[/URL]), and for b=3 this is [URL="https://en.wikipedia.org/wiki/Eisenstein_prime"]Eisenstein prime[/URL] (see [URL="https://oeis.org/A239842"]https://oeis.org/A239842[/URL], [URL="https://oeis.org/A125743"]https://oeis.org/A125743[/URL], [URL="https://oeis.org/A125744"]https://oeis.org/A125744[/URL]), thus this is the [URL="https://en.wikipedia.org/wiki/Cyclotomic_field"]cyclotomic field[/URL] Q(zeta_(2*b)), and for the class numbers, see [URL="https://oeis.org/A061653"]https://oeis.org/A061653[/URL] [URL="https://oeis.org/A000927"]https://oeis.org/A000927[/URL] [URL="https://oeis.org/A055513"]https://oeis.org/A055513[/URL], and thus also related to [URL="https://oeis.org/A000928"]Bernoulli irregular primes[/URL] (primes p dividing Q(zeta_(2*p))) and [URL="https://oeis.org/A120337"]Euler irregular primes[/URL] (primes p dividing Q(zeta_(4*p))/Q(zeta_(2*p))), and there are only 30 natural number b such that Q(zeta_(2*b)) has class number 1: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 30, 33, 35, 42, 45} (the largest such number is 45, and see [URL="https://oeis.org/A018253"]https://oeis.org/A018253[/URL], divisors of 24 have many special properties, including "the only positive integers n with the property m^2 == 1 (mod n) for all integer m coprime to n" "all [URL="https://en.wikipedia.org/wiki/Dirichlet_character"]Dirichlet characters[/URL] are real" "numbers n that are divisible by all numbers less than or equal to the square root of n", and note that 24*45^n+-1 are composites for all small positive integers n, but this is [URL="https://en.wikipedia.org/wiki/Strong_law_of_small_numbers"]strong law of small numbers[/URL], 24*45^18522+1 and 24*45^153355-1 are primes (since they can be proven prime by N-1 and N+1, respectively, they are proven primes, i.e. not just probable primes), and they are the first primes of the form 24*45^n+1 and 24*45^n-1, and thus minimal primes (start with b+1) in base b=45), see [URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119"]list of the multiplicative order of b mod p for b<=128 and primes p<=4096[/URL], [URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339"]list of primes p such that the multiplicative order of b mod p is n for 2<=b<=64 and 1<=n<=64[/URL] (the same lists in factorizations of b^n+-1: [URL="http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=64&FExp=1&TExp=64&c0="]http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=101&FExp=1&TExp=100&c0=[/URL] [URL="http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=101&FExp=1&TExp=100&c0="]http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=101&FExp=1&TExp=100&c0=[/URL] [URL="http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=101&FExp=1&TExp=100&c0=&LM="]http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=64&FExp=1&TExp=64&c0=&LM=[/URL]), [URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039703851"]smallest prime p such that znorder(Mod(m,p)) = (p-1)/n for 2<=m<=128 and 1<=n<=128[/URL] (p is [URL="https://en.wikipedia.org/wiki/Full_reptend_prime#n-th_level_reptend_prime"]n-th level reptend prime[/URL] base m, and if and only if n=1, then m is primitive root mod p, p is full reptend prime base m, and for all r dividing p-1, if and only if n is divisible by r, then m is r-th power residue ([URL="https://en.wikipedia.org/wiki/Quadratic_residue"]quadratic residue[/URL] for r=2, [URL="https://en.wikipedia.org/wiki/Cubic_residue"]cubic residue[/URL] for r=3, [URL="https://en.wikipedia.org/wiki/Quartic_reciprocity"]quartic residue[/URL] for r=4, ..., they are related to [URL="https://en.wikipedia.org/wiki/Power_residue_symbol"]r-th power residue symbol[/URL], a generalization of [URL="https://en.wikipedia.org/wiki/Legendre_symbol"]Legendre symbol[/URL], [URL="https://en.wikipedia.org/wiki/Jacobi_symbol"]Jacobi symbol[/URL], and [URL="https://en.wikipedia.org/wiki/Kronecker_symbol"]Kronecker symbol[/URL], by using [URL="https://en.wikipedia.org/wiki/Dirichlet_character"]Dirichlet character[/URL], which uses r-th [URL="https://en.wikipedia.org/wiki/Root_of_unity"]root of unity[/URL] and is important in [URL="https://en.wikipedia.org/wiki/Dirichlet_L-function"]Dirichlet L-function[/URL] (a generalization of [URL="https://en.wikipedia.org/wiki/Riemann_zeta_function"]Riemann zeta function[/URL]), and hence related to [URL="https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis"]generalized Riemann hypothesis[/URL] and [URL="https://en.wikipedia.org/wiki/Riemann_hypothesis"]Riemann hypothesis[/URL]) mod p), [URL="https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1049313437"]bases b such that Phi(n,b) has algebra factors or small prime factors[/URL], [URL="https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/bases%20b%20such%20that%20there%20is%20unique%20prime%20with%20period%20length%20n"]bases b such that there is unique prime with period length n[/URL], [URL="https://raw.githubusercontent.com/xayahrainie4793/text-file-stored/main/unique%20period%20length%20in%20base%20b"]unique period length in base b[/URL]

(these references only include the multiplicative order of the base (b) mod the primes (i.e. znorder(Mod(b,p)) with prime p), if you want to calculate the multiplicative order of the base (b) mod [I]composite[/I] number c coprime to b, factor c to [URL="https://en.wikipedia.org/wiki/Integer_factorization"]product of distinct prime powers[/URL], and calculated the multiplicative order of b mod p^e (i.e. znorder(Mod(b,p^e))) for all these prime powers p^e, and znorder(Mod(b,p^e)) = p^max(e-r(b,p),0)*znorder(Mod(b,p)), where r(b,p) is the largest integer s such that p^s divides b^(p-1)-1, the primes p such that r(b,p) > 1 are called generalized [URL="https://en.wikipedia.org/wiki/Wieferich_prime"]Wieferich prime[/URL] base b, and if r(p,q) and r(q,p) are both > 1 for primes p and q, then (p,q) are called [URL="https://en.wikipedia.org/wiki/Wieferich_pair"]Wieferich pair[/URL], generalized Wieferich primes and Wieferich pairs are related to [URL="https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem"]Fermat Last Theorem[/URL] and [URL="https://en.wikipedia.org/wiki/Abc_conjecture"]abc conjecture[/URL] and [URL="https://en.wikipedia.org/wiki/Catalan%27s_conjecture"]Catalan conjecture[/URL], and for the values of r(b,p) see [URL="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt"]http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt[/URL] [URL="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt"]http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt[/URL] [URL="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math11/fer_quo.htm"]http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math11/fer_quo.htm[/URL] [URL="http://www.urticator.net/essay/6/624.html"]http://www.urticator.net/essay/6/624.html[/URL] [URL="https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html"]https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html[/URL] (prime bases) [URL="http://download2.polytechnic.edu.na/pub4/sourceforge/w/wi/wieferich/results/table.txt"]http://download2.polytechnic.edu.na/pub4/sourceforge/w/wi/wieferich/results/table.txt[/URL] (broken link) [URL="http://www.sci.kobe-u.ac.jp/old/seminar/pdf/2008_yamazaki.pdf"]http://www.sci.kobe-u.ac.jp/old/seminar/pdf/2008_yamazaki.pdf[/URL], data is available for primes p <= search limit in these two pages, for a base b, if p is not list here then r(b,p) = 1, if p is list here with no exponent given then r(b,p) = 2, if p is list here with an exponent given then r(b,p) = this exponent, [URL="https://en.wikipedia.org/wiki/Perfect_power"]perfect power[/URL] bases are not listed in these two pages, and r(b^m,p) = p^s*r(b,p) if p is odd prime, where s is the largest nonnegative integer such that p^s divides m, r(b^m,2) = largest nonnegative integer s such that 2^s divides b^m-1, finally, calculate the [URL="https://en.wikipedia.org/wiki/Least_common_multiple"]least common multiple[/URL] of these multiplicative orders of b mod p^e) (related links: [URL="http://go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm"]http://go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm[/URL] [URL="http://go.helms-net.de/math/expdioph/fermatquotients.pdf"]http://go.helms-net.de/math/expdioph/fermatquotients.pdf[/URL])

This is the data for known generalized Wieferich primes to bases 2<=b<=36 (prime appear more times means this prime is higher-order Wieferich prime to this base, i.e. prime p appear n times means p^(n+1) | b^(p-1)-1) (I stop at base 36 since this is the largest base whose digits can be represented using the 10 [URL="https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numerals"]Arabic numerals[/URL] and the 26 [URL="https://en.wikipedia.org/wiki/Latin_alphabet"]Latin letters[/URL])

[CODE]
2: 1093, 3511
3: 11, 1006003
4: 1093, 3511
5: 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
6: 66161, 534851, 3152573
7: 5, 491531
8: 3, 1093, 3511
9: 2, 2, 11, 1006003
10: 3, 487, 56598313
11: 71
12: 2693, 123653
13: 2, 863, 1747591
14: 29, 353, 7596952219
15: 29131, 119327070011
16: 1093, 3511
17: 2, 2, 2, 3, 46021, 48947, 478225523351
18: 5, 7, 7, 37, 331, 33923, 1284043
19: 3, 7, 7, 13, 43, 137, 63061489
20: 281, 46457, 9377747, 122959073
21: 2
22: 13, 673, 1595813, 492366587, 9809862296159
23: 13, 2481757, 13703077, 15546404183, 2549536629329
24: 5, 25633
25: 2, 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
26: 3, 3, 5, 71, 486999673, 6695256707
27: 11, 1006003
28: 3, 3, 19, 23
29: 2
30: 7, 160541, 94727075783
31: 7, 79, 6451, 2806861
32: 5, 1093, 3511
33: 2, 2, 2, 2, 233, 47441, 9639595369
34: 46145917691
35: 3, 1613, 3571
36: 66161, 534851, 3152573
[/CODE]

(the generalized Wieferich primes are more important for [URL="https://en.wikipedia.org/wiki/Square-free_integer"]squarefree[/URL] bases b (and because of [URL="https://en.wikipedia.org/wiki/Wieferich_pair"]Wieferich pairs[/URL] related to [URL="https://en.wikipedia.org/wiki/Abc_conjecture"]abc conjecture[/URL] and [URL="https://en.wikipedia.org/wiki/Catalan%27s_conjecture"]Catalan conjecture[/URL], the generalized Wieferich primes are much more important for prime bases b), since for squarefree bases b, the [URL="https://en.wikipedia.org/wiki/Ring_of_integers"]ring of integers[/URL] of Q(b^(1/p)) is not Z[b^(1/p)] if and only if p is generalized Wieferich prime base b, and for non-squarefree bases b, the [URL="https://en.wikipedia.org/wiki/Ring_of_integers"]ring of integers[/URL] of Q(b^(1/p)) is [I]always[/I] not Z[b^(1/p)], references: [URL="https://oeis.org/A342390"]https://oeis.org/A342390[/URL] [URL="https://oeis.org/A342391"]https://oeis.org/A342391[/URL] [URL="https://oeis.org/A342392"]https://oeis.org/A342392[/URL] [URL="https://oeis.org/A342393"]https://oeis.org/A342393[/URL])

By [URL="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem"]Fermat's little theorem[/URL], multiplicative order of the b mod prime p (i.e. znorder(Mod(b,p)) with prime p) always divide p-1, however, if [I]composite[/I] c [URL="https://en.wikipedia.org/wiki/Coprime_integers"]coprime[/URL] to b also satisfies that multiplicative order of the b mod c (i.e. znorder(Mod(b,c))) divides p-1, then c is called [URL="https://en.wikipedia.org/wiki/Fermat_pseudoprime"]Fermat pseudoprime[/URL] base b, it is known that for every base b>=2 there are infinitely many Fermat pseudoprimes base b (see theorem 1 in [URL="https://math.dartmouth.edu/~carlp/PDF/paper25.pdf"]https://math.dartmouth.edu/~carlp/PDF/paper25.pdf[/URL], we can use the factors of b^n-1 and b^n+1 to easily get these numbers), also, by [URL="https://en.wikipedia.org/wiki/Euler%27s_theorem"]Euler theorem[/URL], multiplicative order of the b mod any number n (i.e. znorder(Mod(b,n))) always divide [URL="https://en.wikipedia.org/wiki/Euler%27s_totient_function"]eulerphi[/URL](n), in fact, always divide [URL="https://en.wikipedia.org/wiki/Carmichael_function"]carmichaellambda[/URL](n), and carmichaellambda(n) is a factor of eulerphi(n) for all positive integer n, if and only if they are equal, then n has primitive roots, such n are listed in [URL="https://oeis.org/A033948"]https://oeis.org/A033948[/URL]

(references for factoring Cunningham numbers (factorization of b^n+-1, which is equivalent to factorization of Zs(n,b,1)): [URL="https://homes.cerias.purdue.edu/~ssw/cun/index.html"]b<=12[/URL] [URL="https://maths-people.anu.edu.au/~brent/factors.html"]13<=b<=99[/URL] [URL="https://stdkmd.net/nrr/repunit/"]b=10[/URL] [URL="https://homes.cerias.purdue.edu/~ssw/bell/index.html"]b=n and b is prime[/URL] [URL="http://myfactors.mooo.com/"]any b[/URL] [URL="http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm"]any b[/URL], also see [URL="https://stdkmd.net/nrr/repunit/repunitnote.htm"]this page[/URL] and [URL="https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980"]this page[/URL] and [URL="https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf"]this page[/URL] and [URL="http://web.archive.org/web/20170515153924/http://bitc.bme.emory.edu/~lzhou/blogs/?p=263"]this page[/URL])

related to "non-generous primes": [URL="https://oeis.org/A055578"]https://oeis.org/A055578[/URL] [URL="https://oeis.org/A101710"]https://oeis.org/A101710[/URL]

Related OEIS sequences:

znorder: [URL="https://oeis.org/A250211"]A250211[/URL] [URL="https://oeis.org/A139366"]A139366[/URL] [URL="https://oeis.org/A057593"]A057593[/URL] [URL="https://oeis.org/A002326"]A002326[/URL] [URL="https://oeis.org/A014664"]A014664[/URL] [URL="https://oeis.org/A062117"]A062117[/URL] [URL="https://oeis.org/A003571"]A003571[/URL] [URL="https://oeis.org/A003572"]A003572[/URL] [URL="https://oeis.org/A082654"]A082654[/URL] [URL="https://oeis.org/A211241"]A211241[/URL] [URL="https://oeis.org/A211242"]A211242[/URL] [URL="https://oeis.org/A007732"]A007732[/URL] [URL="https://oeis.org/A051626"]A051626[/URL] [URL="https://oeis.org/A006556"]A006556[/URL] [URL="https://oeis.org/A002371"]A002371[/URL] [URL="https://oeis.org/A001917"]A001917[/URL] [URL="https://oeis.org/A054471"]A054471[/URL]

cyclotomic numbers: [URL="https://oeis.org/A253240"]A253240[/URL] [URL="https://oeis.org/A019320"]A019320[/URL] [URL="https://oeis.org/A019321"]A019321[/URL] [URL="https://oeis.org/A019322"]A019322[/URL] [URL="https://oeis.org/A019322"]A019323[/URL] [URL="https://oeis.org/A019324"]A019324[/URL] [URL="https://oeis.org/A019325"]A019325[/URL] [URL="https://oeis.org/A019326"]A019326[/URL] [URL="https://oeis.org/A019327"]A019327[/URL] [URL="https://oeis.org/A019328"]A019328[/URL] [URL="https://oeis.org/A019329"]A019329[/URL] [URL="https://oeis.org/A019330"]A019330[/URL] [URL="https://oeis.org/A019331"]A019331[/URL]

Zsigmondy numbers: [URL="https://oeis.org/A323748"]A323748[/URL] [URL="https://oeis.org/A064078"]A064078[/URL] [URL="https://oeis.org/A064079"]A064079[/URL] [URL="https://oeis.org/A064080"]A064080[/URL] [URL="https://oeis.org/A064081"]A064081[/URL] [URL="https://oeis.org/A064082"]A064082[/URL] [URL="https://oeis.org/A064083"]A064083[/URL]

cyclotomic number is prime: [URL="https://oeis.org/A085398"]A085398[/URL] [URL="https://oeis.org/A117544"]A117544[/URL] [URL="https://oeis.org/A117545"]A117545[/URL] [URL="https://oeis.org/A066180"]A066180[/URL] [URL="https://oeis.org/A103795"]A103795[/URL] [URL="https://oeis.org/A056993"]A056993[/URL] [URL="https://oeis.org/A153438"]A153438[/URL] [URL="https://oeis.org/A246120"]A246120[/URL] [URL="https://oeis.org/A246119"]A246119[/URL] [URL="https://oeis.org/A298206"]A298206[/URL] [URL="https://oeis.org/A246121"]A246121[/URL] [URL="https://oeis.org/A206418"]A206418[/URL] [URL="https://oeis.org/A205506"]A205506[/URL] [URL="https://oeis.org/A181980"]A181980[/URL] [URL="https://oeis.org/A072226"]A072226[/URL] [URL="https://oeis.org/A138933"]A138933[/URL] [URL="https://oeis.org/A138934"]A138934[/URL] [URL="https://oeis.org/A138935"]A138935[/URL] [URL="https://oeis.org/A138936"]A138936[/URL] [URL="https://oeis.org/A138937"]A138937[/URL] [URL="https://oeis.org/A138938"]A138938[/URL] [URL="https://oeis.org/A138939"]A138939[/URL] [URL="https://oeis.org/A138940"]A138940[/URL]

Zsigmondy number is prime: [URL="https://oeis.org/A275530"]A275530[/URL] [URL="https://oeis.org/A161508"]A161508[/URL] [URL="https://oeis.org/A247071"]A247071[/URL] [URL="https://oeis.org/A007498"]A007498[/URL] [URL="https://oeis.org/A051627"]A051627[/URL]

primitive root: [URL="https://oeis.org/A001122"]A001122[/URL] [URL="https://oeis.org/A019334"]A019334[/URL] [URL="https://oeis.org/A019335"]A019335[/URL] [URL="https://oeis.org/A019336"]A019336[/URL] [URL="https://oeis.org/A019337"]A019337[/URL] [URL="https://oeis.org/A019338"]A019338[/URL] [URL="https://oeis.org/A001913"]A001913[/URL] [URL="https://oeis.org/A019339"]A019339[/URL] [URL="https://oeis.org/A019340"]A019340[/URL] [URL="https://oeis.org/A001918"]A001918[/URL] [URL="https://oeis.org/A046145"]A046145[/URL] [URL="https://oeis.org/A103309"]A103309[/URL] [URL="https://oeis.org/A122028"]A122028[/URL] [URL="https://oeis.org/A060749"]A060749[/URL] [URL="https://oeis.org/A046147"]A046147[/URL] [URL="https://oeis.org/A023048"]A023048[/URL] [URL="https://oeis.org/A133433"]A133433[/URL] [URL="https://oeis.org/A133432"]A133432[/URL] [URL="https://oeis.org/A214158"]A214158[/URL] [URL="https://oeis.org/A002230"]A002230[/URL] [URL="https://oeis.org/A002229"]A002229[/URL] [URL="https://oeis.org/A262264"]A262264[/URL] [URL="https://oeis.org/A056619"]A056619[/URL] [URL="https://oeis.org/A023049"]A023049[/URL] [URL="https://oeis.org/A280015"]A280015[/URL]

Fermat pseudoprime: [URL="https://oeis.org/A001567"]A001567[/URL] [URL="https://oeis.org/A005935"]A005935[/URL] [URL="https://oeis.org/A090086"]A090086[/URL] [URL="https://oeis.org/A007535"]A007535[/URL] [URL="https://oeis.org/A000790"]A000790[/URL] [URL="https://oeis.org/A239293"]A239293[/URL] [URL="https://oeis.org/A090087"]A090087[/URL] [URL="https://oeis.org/A090085"]A090085[/URL] [URL="https://oeis.org/A063994"]A063994[/URL] [URL="https://oeis.org/A064234"]A064234[/URL] [URL="https://oeis.org/A247074"]A247074[/URL] [URL="https://oeis.org/A181780"]A181780[/URL] [URL="https://oeis.org/A211455"]A211455[/URL] [URL="https://oeis.org/A211458"]A211458[/URL]

Wieferich prime: [URL="https://oeis.org/A039951"]A039951[/URL] [URL="https://oeis.org/A174422"]A174422[/URL] [URL="https://oeis.org/A096082"]A096082[/URL] [URL="https://oeis.org/A247072"]A247072[/URL] [URL="https://oeis.org/A268352"]A268352[/URL] [URL="https://oeis.org/A178871"]A178871[/URL] [URL="https://oeis.org/A001220"]A001220[/URL] [URL="https://oeis.org/A014127"]A014127[/URL] [URL="https://oeis.org/A123692"]A123692[/URL] [URL="https://oeis.org/A212583"]A212583[/URL] [URL="https://oeis.org/A123693"]A123693[/URL] [URL="https://oeis.org/A045616"]A045616[/URL] [URL="https://oeis.org/A111027"]A111027[/URL] [URL="https://oeis.org/A039678"]A039678[/URL] [URL="https://oeis.org/A143548"]A143548[/URL]

sweety439 2021-06-29 16:29

README for the file [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL]:

* 12{3}45 means family {1245, 12345, 123345, 1233345, 12333345, 123333345, ...}

* A means digit value 10, B means digit value 11, C means digit value 12, ...

* z means digit value base-1, y means digit value base-2, x means digit value base-3, w means digit value base-4, ...

* the numbers in the list is the length of the smallest primes or PRPs in this family in this base (only count numbers > base) (e.g. family 3{z} in base 72, the smallest prime is 4*72^1119849-1, which has 1119850 digits in base 72, thus the number for family 3{z} for base 72 is 1119850)

* NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base)

* RC: this family can be proven to only contain composite numbers (only count numbers > base)

* unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base)

Test limit of the length of the families for the file [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL] (for bases 2<=b<=1024):

1{0}1: >=8388608 (reference: [URL="http://www.noprimeleftbehind.net/crus/GFN-primes.htm"]http://www.noprimeleftbehind.net/crus/GFN-primes.htm[/URL], also see [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL])

{1}: >=100000 (thanks to Michael Stocker, reference: [URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-1%29%2Fa&action=Search"]http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-1%29%2Fa&action=Search[/URL], also see [URL="https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html"]https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html[/URL] and [URL="http://www.fermatquotient.com/PrimSerien/GenRepu.txt"]http://www.fermatquotient.com/PrimSerien/GenRepu.txt[/URL])

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1, 1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}: >=100000 (reference: [URL="https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n[/URL], [URL="https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n"]https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n[/URL])

1{z}: >=200000 (reference: [URL="https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217"]https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217[/URL])

z{0}1: >=100000 (reference: [URL="https://www.rieselprime.de/ziki/Williams_prime_MP_least"]https://www.rieselprime.de/ziki/Williams_prime_MP_least[/URL])

y{z}: >=200000 (reference: [URL="https://www.rieselprime.de/ziki/Williams_prime_MM_least"]https://www.rieselprime.de/ziki/Williams_prime_MM_least[/URL], also see [URL="https://harvey563.tripod.com/wills.txt"]https://harvey563.tripod.com/wills.txt[/URL])

1{0}2, {z}y, 1{0}z, {z}1, {y}z: >=5000 (by me)

other families: >=2500 (by me)

sweety439 2021-07-31 03:54

There is someone else who also exclude the single-digit primes, but his research is about [URL="https://en.wikipedia.org/wiki/Substring"]substring[/URL] instead of [URL="https://en.wikipedia.org/wiki/Subsequence"]subsequence[/URL], see [URL="https://www.mersenneforum.org/showpost.php?p=235383&postcount=42"]this post[/URL]

sweety439 2021-07-31 08:35

Related search for minimal primes (generalized form: (a*b^n+c)/d) in [URL="http://www.primenumbers.net/prptop/prptop.php"]top 10000 probable primes[/URL]:

[URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En%2Bc&action=Search"]b^n+c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=b%5En-c&action=Search"]b^n-c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En%2Bc&action=Search"]a*b^n+c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=a*b%5En-c&action=Search"]a*b^n-c[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En%2Bc%29%2Fd&action=Search"](b^n+c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28b%5En-c%29%2Fd&action=Search"](b^n-c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En%2Bc%29%2Fd&action=Search"](a*b^n+c)/d[/URL]

[URL="http://www.primenumbers.net/prptop/searchform.php?form=%28a*b%5En-c%29%2Fd&action=Search"](a*b^n-c)/d[/URL]

Also for the special case c = +-1 and d = 1, they are [I]proven[/I] primes (i.e. not merely probable primes), the search page in [URL="https://primes.utm.edu/primes/"]top 5000 proven primes[/URL]:

[URL="https://primes.utm.edu/primes/search_proth.php"]https://primes.utm.edu/primes/search_proth.php[/URL]

[URL="https://primes.utm.edu/primes/search.php?Description=%5E[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML"]b^n+1[/URL]

[URL="https://primes.utm.edu/primes/search.php?Description=%5E[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML"]b^n-1[/URL]

[URL="https://primes.utm.edu/primes/search.php?Description=[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D%2B1&OnList=all&Number=1000000&Style=HTML"]a*b^n+1[/URL]

[URL="https://primes.utm.edu/primes/search.php?Description=[[:digit:]]%7B1,%7D*[[:digit:]]%7B1,%7D%5E[[:digit:]]%7B1,%7D-1&OnList=all&Number=1000000&Style=HTML"]a*b^n-1[/URL]

sweety439 2021-07-31 08:53

1 Attachment(s)
Update newest data for minimal primes (start with b+1) in bases 2<=b<=16

Newest condensed table:

[CODE]
b number of quasi-minimal primes base b base-b form of largest known quasi-minimal prime base b length of largest known quasi-minimal prime base b algebraic ((a×bn+c)/d) form of largest known quasi-minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 ≥71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
9 ≥149 3(0^1158)11 1161 3×9^1160+10
10 77 5(0^28)27 31 5×10^30+27
11 ≥914 55(7^1011) 1013 (607×11^1011−7)/10
12 106 4(0^39)77 42 4×12^41+91
13 ≥2492 8(0^32017)111 32021 8×13^32020+183
14 ≥605 4(D^19698) 19699 5×14^19698−1
15 ≥1171 (7^155)97 157 (15^157+59)/2
16 ≥1991 D(B^32234) 32235 (206×16^32234−11)/15
[/CODE]

There are three unsolved families known to me:

[CODE]
Base 11: 57*
Base 13: 95*
Base 13: A3*A
[/CODE]

sweety439 2021-08-07 06:46

[URL="https://sites.google.com/view/minimal--primes"]https://sites.google.com/view/minimal--primes[/URL]

sweety439 2021-08-08 14:50

1 Attachment(s)
Update the data text file.

sweety439 2021-08-08 16:58

[QUOTE=sweety439;568922]Now, we proved the set of minimal primes (start with b+1, which is equivalent to start with b, if b is composite) of base b=12:

[CODE]
11
15
17
1B
25
27
31
35
37
3B
45
4B
51
57
5B
61
67
6B
75
81
85
87
8B
91
95
A7
AB
B5
B7
221
241
2A1
2B1
2BB
401
421
447
471
497
565
655
665
701
70B
721
747
771
77B
797
7A1
7BB
907
90B
9BB
A41
B21
B2B
2001
200B
202B
222B
229B
292B
299B
4441
4707
4777
6A05
6AA5
729B
7441
7B41
929B
9777
992B
9947
997B
9997
A0A1
A201
A605
A6A5
AA65
B001
B0B1
BB01
BB41
600A5
7999B
9999B
AAAA1
B04A1
B0B9B
BAA01
BAAA1
BB09B
BBBB1
44AAA1
A00065
BBBAA1
AAA0001
B00099B
AA000001
BBBBBB99B
B0000000000000000000000000009B
400000000000000000000000000000000000000077
[/CODE][/QUOTE]

Now I try to prove base 7:

In base 7, the possible (first digit,last digit) for an element with >=3 digits in the minimal set of the strings for primes with at least two digits are

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

* Case (1,1):

** Since 14, 16, 41, 61, [B]131[/B] are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes)

*** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {}

**** If there are >=3 number of 1 and 5 in the {}:

***** If there is 111 in the {}, then we have the prime [B]11111[/B]

***** If there is 115 in the {}, then the prime 115 is a subsequence

***** If there is 151 in the {}, then the prime 115 is a subsequence

***** If there is 155 in the {}, then the prime 155 is a subsequence

***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime

***** If there is 515 in the {}, then the prime 515 is a subsequence

***** If there is 551 in the {}, then the prime 551 is a subsequence

***** If there is 555 in the {}, then the prime 551 is a subsequence

**** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1

***** For the 1{0,2}1{0,2}1 family, since [B]1211[/B] is prime, we only need to consider the family 1{0}1{0,2}1

****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1

******* Since [B]11201[/B] is prime, we only need to consider the family 1{0}1{0}21

******** The smallest prime of the form 11{0}21 is [B]1100021[/B]

******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime

******** The smallest prime of the form 1001{0}21 is [B]100121[/B]

********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families

***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1

****** Since [B]1051[/B] is prime, we only need to consider the family 15{0}1

******* The smallest prime of the form 15{0}1 is [B]150001[/B]

sweety439 2021-08-08 17:29

* Case (1,2):

** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes)

*** Since [B]1112[/B] and [B]1222[/B] are primes, there is at most one 1 and at most one 2 in {}

**** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime.

**** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime.

**** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2

***** Since [B]1022[/B] and [B]1202[/B] are primes, we only need to consider the number 122

****** 122 is not prime.

**** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime.

* Case (1,3):

** Since 14, 16, 23, 43, [B]113[/B], [B]133[/B] are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes)

*** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3

**** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime.

**** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime.

* Case (1,4):

** [B]14[/B] is prime, and thus the only minimal prime in this family.

* Case (1,5):

** Since 14, 16, 25, 65, [B]115[/B], [B]155[/B] are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes)

*** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime.

* Case (1,6):

** [B]16[/B] is prime, and thus the only minimal prime in this family.

sweety439 2021-08-08 18:10

* Case (2,1):

** Since 23, 25, 41, 61, [B]221[/B] are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes)

*** Since [B]2111[/B] is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1

**** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime.

**** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime.

* Case (2,2):

** Since 23, 25, 32, 52, [B]212[/B] are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime.

* Case (2,3):

** [B]23[/B] is prime, and thus the only minimal prime in this family.

* Case (2,4):

** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime.

* Case (2,5):

** [B]25[/B] is prime, and thus the only minimal prime in this family.

* Case (2,6):

** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes)

*** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime.

sweety439 2021-08-09 10:03

[QUOTE=sweety439;566057]* Case (6,1):

** Since 65, 21, 51, [B]631[/B], [B]661[/B] are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Since 111, 141, 401, 471, 701, 711, [B]6101[/B], [B]6441[/B] are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes)

**** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes)

***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, [B]60411[/B] are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes)

****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, [B]60741[/B] are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes)

***** Since 471, [B]60171[/B] is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes)

****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime.

**** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11

***** For the 6{0}1{7}1 family, since [B]60171[/B] is prime, and thus the only minimal prime in the family 6{0}1{7}1.

***** For the 6{0,4}11 family, since 401, 6441, [B]60411[/B] are primes, we only need to consider the number 6411 and the family 6{0}11

****** 6411 is not prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

**** For the 6{0,7}4{1}1 family, since [B]60411[/B] is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41

***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411

****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes.

***** For the 6{0,7}41 family, since 701, 6777, [B]60741[/B] are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes)

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

****** Neither of 6741, 67741 are primes.

***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes)

****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes)

******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71

******** Since 717 and [B]60171[/B] are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes)

********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes)

********** 6171 is not prime.

****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes.[/QUOTE]

A simpler proof for base 8 case (6,1):

** Since 65, 21, 51, [B]631[/B], [B]661[/B] are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Numbers containing 4: (note that the number cannot contain two or more 4's, or [B]6441[/B] will be a subsequence)

**** The form is 6{0,1,7}4{0,1,7}1

***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1

****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411

******* For the 6{0,7}41 family, since [B]60741[/B] is prime, we only need to consider the family 6{7}{0}41

******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41

********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime.

********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime.

******* For the 6{0,7}411 family, since [B]60411[/B] is prime, we only need to consider the family 6{7}411

******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime)

*** Numbers not containing 4:

**** The form is 6{0,1,7}1

***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1

****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime.

****** For the 6{0,7}1{0,7}1 family, since 711 and [B]6101[/B] are primes, we only need to consider the family 6{0}1{7}1

******* Since [B]60171[/B] is prime, we only need to consider the families 6{0}11 and 61{7}1

******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime)

sweety439 2021-08-16 16:25

2 Attachment(s)
Running bases 17 and 18, these are small minimal primes (start with b+1) up to certain limit in bases b=17 and b=18

More known minimal primes or PRP (start with b+1):

[URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.17.txt"]base 17, not contain single-digit primes nor contain string "10"[/URL]
[URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.18.txt"]base 18, not contain single-digit primes (only three primes: CCCCCCCC1, E0CCCCCC1, GG0000000000000000000000000000001, all other primes are already in the list)[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/non-single-digit-primes/main/smallest%20generalized%20near-repdigit%20prime.txt"]smallest primes of the form {x}y and x{y} for fixed base and fixed digits x and y, all except whose repeating digit (i.e. x for {x}y, y for x{y}) is 1 are minimal primes (start with b+1)[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/non-single-digit-primes/main/smallest%20prime%20of%20the%20form%20x000000y.txt"]smallest primes of the form x{0}y for fixed base and fixed digits x and y, all are minimal primes (start with b+1)[/URL]
[URL="https://raw.githubusercontent.com/xayahrainie4793/non-single-digit-primes/main/x0000yz%20and%20xy0000z.txt"]smallest primes of the form xy{0}z and x{0}yz for fixed base and fixed digits x, y, and z, all are minimal primes (start with b+1)[/URL]
1(0^9019)1F in base 17 (= 17^9021+32)
F7(0^186767)1 in base 17 (= 262*17^186768+1)
97(0^166047)1 in base 17 (= 160*17^166048+1)
57(0^51310)1 in base 17 (= 92*17^51311+1)

sweety439 2021-08-20 07:52

Reserving the two unsolved families in base 16 found by me: {3}AF and {4}DD, the formulas of them are (16^n+619)/5 and (4*16^n+2291)/15, respectively, since the base is even (and hence the divisors are odd), the [I]srsieve[/I] program can be used (we sieve the sequence 16^n+619 and 4*16^n+2291, start with the prime 7), unlike the base 11 unsolved family 5{7} (the formula is (57*11^n-7)/10) and the base 13 unsolved family 9{5} (the formula is (113*13^n-5)/12), which cannot be sieved with [I]srsieve[/I] since (if we sieve the sequences 57*11^n-7 and 113*13^n-5) it will return: "error: all numbers are divisible by 2".

References of [I]srsieve[/I] reserving of the unsolved families for the original minimal prime problem (i.e. the restriction of prime>base is not required) for bases b<=28: [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/extdivisor/data/sieve-nodenom.abc"]sequences a*b^n+c[/URL] [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/extdivisor/data/sieve.abc"]sequences (a*b^n+c)/d with d>1[/URL], the author updates the [I]srsieve[/I] program to make it can sieve the sequence a*b^n+c with a, b, c all odd (i.e. the divisor is even).

sweety439 2021-08-22 11:30

1 Attachment(s)
Newest pdf files attached. Especially, base 16 all primes with <=9 digits are searched.

sweety439 2021-08-23 13:18

1 Attachment(s)
Newest data file attached, for bases 2<=b<=18 (base 17 and 18 only include small minimal primes (start with b+1), not include known large minimal primes (start with b+1) such as 2*17^47+1 = 2(0^46)1 base 17 and the primes in [URL="https://github.com/curtisbright/mepn-data/blob/master/data/minimal.17.txt"]https://github.com/curtisbright/mepn-data/blob/master/data/minimal.17.txt[/URL] and/or [URL="https://github.com/xayahrainie4793/non-single-digit-primes/blob/main/smallest%20generalized%20near-repdigit%20prime.txt"]https://github.com/xayahrainie4793/non-single-digit-primes/blob/main/smallest%20generalized%20near-repdigit%20prime.txt[/URL] for base 17)

sweety439 2021-08-26 16:27

Newest condensed table for bases b<=16:

[CODE]
b number of minimal primes (start with b+1) base b base-b form of largest known minimal prime (start with b+1) base b length of largest known minimal prime (start with b+1) base b algebraic ((a×bn+c)/d) form of largest known minimal prime (start with b+1) base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 ≥71 (3^16)1 17 (7^17−5)/2
8 75 (4^220)7 221 (4×8^221+17)/7
9 ≥149 3(0^1158)11 1161 3×9^1160+10
10 77 5(0^28)27 31 5×10^30+27
11 ≥914 55(7^1011) 1013 (607×11^1011−7)/10
12 106 4(0^39)77 42 4×12^41+91
13 ≥2492 8(0^32017)111 32021 8×13^32020+183
14 ≥605 4(D^19698) 19699 5×14^19698−1
15 ≥1171 (7^155)97 157 (15^157+59)/2
16 ≥2050 D(B^32234) 32235 (206×16^32234−11)/15
[/CODE]

There are five unsolved families for bases b<=16 known to me:

[CODE]
Base 11: 57*
Base 13: 95*
Base 13: A3*A
Base 16: 3*AF
Base 16: 4*DD
[/CODE]

sweety439 2021-08-30 20:10

We have properties for bases 2<=b<=1024:

* Primes with [URL="https://en.wikipedia.org/wiki/Repeating_decimal"]period length[/URL] 1 in this base
* Primes with period length 2 in this base
* Primes with period length 3 in this base
* Primes with period length 4 in this base
* Primes with period length 5 in this base
* Primes with period length 6 in this base
* Primes with period length 7 in this base
* Primes with period length 8 in this base
* Primes with period length 9 in this base
* Primes with period length 10 in this base
* Primes with period length 11 in this base
* Primes with period length 12 in this base
* Primes with period length 13 in this base
* Primes with period length 14 in this base
* Primes with period length 15 in this base
* Primes with period length 16 in this base
* Primes with period length 17 in this base
* Primes with period length 18 in this base
* Primes with period length 19 in this base
* Primes with period length 20 in this base
* Primes with period length 21 in this base
* Primes with period length 22 in this base
* Primes with period length 23 in this base
* Primes with period length 24 in this base
* Primes with period length 25 in this base
* Primes with period length 26 in this base
* Primes with period length 27 in this base
* Primes with period length 28 in this base
* Primes with period length 29 in this base
* Primes with period length 30 in this base
* Primes with period length 31 in this base
* Primes with period length 32 in this base
* Primes with period length 33 in this base
* Primes with period length 34 in this base
* Primes with period length 35 in this base
* Primes with period length 36 in this base
* Primes with period length 37 in this base
* Primes with period length 38 in this base
* Primes with period length 39 in this base
* Primes with period length 40 in this base
* Primes with period length 41 in this base
* Primes with period length 42 in this base
* Primes with period length 43 in this base
* Primes with period length 44 in this base
* Primes with period length 45 in this base
* Primes with period length 46 in this base
* Primes with period length 47 in this base
* Primes with period length 48 in this base
* Primes with period length 49 in this base
* Primes with period length 50 in this base
* Primes with period length 51 in this base
* Primes with period length 52 in this base
* Primes with period length 53 in this base
* Primes with period length 54 in this base
* Primes with period length 55 in this base
* Primes with period length 56 in this base
* Primes with period length 57 in this base
* Primes with period length 58 in this base
* Primes with period length 59 in this base
* Primes with period length 60 in this base
* Primes with period length 61 in this base
* Primes with period length 62 in this base
* Primes with period length 63 in this base
* Primes with period length 64 in this base

* Known [URL="https://en.wikipedia.org/wiki/Unique_prime"]unique period lengths[/URL] in this base
* [URL="https://en.wikipedia.org/wiki/Multiplicative_order"]znorder[/URL]([URL="https://en.wikipedia.org/wiki/Modular_arithmetic"]Mod[/URL](this base,p)) for all primes p < 2^16 not dividing this base
* The smallest prime p such that znorder(Mod((this base,p)) = (p-1)/n for given number 1<=n<=64 (if impossible, then write "impossible")
* Known generalized [URL="https://en.wikipedia.org/wiki/Wieferich_prime"]Wieferich primes[/URL] in this base

* The CK and status for [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm"]Sierpinski conjecture[/URL] base b
* The CK and status for [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm"]Riesel conjecture[/URL] base b
* The CK and status for [URL="https://docs.google.com/document/d/e/2PACX-1vSQlPrWZgVM1g5spyMs_USkKy3XEGcBsadeLc82JmQVbXCOWbbcSkuHMtO_EmspQME3ITGNvoCcffZt/pub"]extended Sierpinski conjecture[/URL] base b
* The CK and status for [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended Riesel conjecture[/URL] base b

(if these forms have [I]no possible[/I] primes > base, then write "proven to contain no primes > base")

* Known lengths of primes of the form 1{0}1
* Known lengths of (probable) primes of the form 1{0}2
* Known lengths of (probable) primes of the form 1{0}3
* Known lengths of (probable) primes of the form 1{0}4
* Known lengths of (probable) primes of the form 1{0}z
* Known lengths of (probable) primes of the form {1}
* Known lengths of (probable) primes of the form 1{2}
* Known lengths of (probable) primes of the form 1{3}
* Known lengths of (probable) primes of the form 1{4}
* Known lengths of primes of the form 1{z}
* Known lengths of primes of the form 2{0}1
* Known lengths of (probable) primes of the form 2{0}3
* Known lengths of primes of the form 2{z}
* Known lengths of primes of the form 3{0}1
* Known lengths of (probable) primes of the form 3{0}2
* Known lengths of (probable) primes of the form 3{0}4
* Known lengths of primes of the form 3{z}
* Known lengths of primes of the form 4{0}1
* Known lengths of (probable) primes of the form 4{0}3
* Known lengths of primes of the form 4{z}
* Known lengths of primes of the form 5{0}1
* Known lengths of primes of the form 5{z}
* Known lengths of primes of the form 6{0}1
* Known lengths of primes of the form 6{z}
* Known lengths of primes of the form 7{0}1
* Known lengths of primes of the form 7{z}
* Known lengths of primes of the form 8{0}1
* Known lengths of primes of the form 8{z}
* Known lengths of primes of the form 9{0}1
* Known lengths of primes of the form 9{z}
* Known lengths of primes of the form A{0}1
* Known lengths of primes of the form A{z}
* Known lengths of primes of the form B{0}1
* Known lengths of primes of the form B{z}
* Known lengths of primes of the form C{0}1
* Known lengths of (probable) primes of the form {#}$ (for odd bases b, # = (b-1)/2, $ = (b+1)/2)
* Known lengths of (probable) primes of the form {y}z
* Known lengths of primes of the form y{z}
* Known lengths of primes of the form z{0}1
* Known lengths of (probable) primes of the form {z}1
* Known lengths of (probable) primes of the form {z}w
* Known lengths of (probable) primes of the form {z}x
* Known lengths of (probable) primes of the form {z}y

sweety439 2021-08-31 21:39

For all n>=1, all minimal primes (start with b+1) base b which are > b^n are also minimal primes (start with b'+1) base b' = b^n, e.g. all minimal primes (start with b+1) base b = 10 which are > 10^n are also minimal primes (start with b'+1) base b' = 10^n

Thus, for all n>=1, the largest minimal prime (start with b'+1) base b' = b^n is always >= the largest minimal prime (start with b+1) base b, besides, if base b is not solved, then for all n>=1, base b^n are also not solved.

sweety439 2021-09-05 20:21

Known minimal primes (start with b+1) in base b=17:

Small ones: (written in base b)

[CODE]
12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 700D5, 700DB, 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991999999, 995555555, 9955FFFFB, 995FFFFFF, 997770705, 998858888, 999000059, 999020055, 99909C555, 99990C555, 9999585B8, 999985888, 99999222G, 999992556, 999995FFF, 999996222, 999999902, 99999992G, 999999B22, 99999C9CB, 9999GCFFF, 9999GFFFC, 999F77777, 9C8888888, 9C9999918, 9CCC666C2, 9CCCC6206, 9CCCCCC62, 9CD000DDD, 9D00DDDD5, 9D7777773, 9DDDD1009, 9DDDDDDD5, 9GGGGGG2G, A00000108, A0000051F, A0000056E, A00000A85, A000010F1, A0005100F, A00501FFF, A00555552, A0EAAA555, A0F000FFC, A0FFFFF4A, A25555555, A41444444, A55555255, A5AAAA55E, A6000000F, A6660666F, A70000101, A74444444, A77777774, ...
[/CODE]

Found by original minimal primes search: (written in base b)

See [URL="https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.17.txt"]https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/minimal.17.txt[/URL] (data is too large to post here)

sweety439 2021-09-05 20:23

Found by smallest generalized near-repdigit primes (i.e. of the form x{y} or {x}y) base b: (written in base 10) (the numbers whose repeating digit (i.e. y for x{y}, or x for {x}y) is 1 are not minimal primes (start with b+1), but still post here)

[CODE]
17, {1}, 1: 307
17, 1, {1}: 307
17, {1}, 2: 19
17, 1, {2}: 19
17, 1, {3}: 99181
17, 1, {4}: 83916100750126603685919194992742897025955933234826458732651955310888606001
17, {1}, 5: 311
17, 1, {5}: 379
17, {1}, 6: 23
17, 1, {6}: 23
17, {1}, 7: 313
17, 1, {7}: 0
17, {1}, 8: 5227
17, 1, {8}: 433
17, 1, {9}: 0
17, 1, {10}: 135721
17, {1}, 11: 317
17, 1, {11}: 487
17, {1}, 12: 29
17, 1, {12}: 29
17, {1}, 13: 0
17, 1, {13}: 523
17, {1}, 14: 31
17, 1, {14}: 31
17, 1, {15}: 0
17, 1, {16}: 577
17, {2}, 1: 613
17, 2, {1}: 10133
17, {2}, 3: 37
17, 2, {3}: 37
17, {2}, 5: 617
17, {2}, 7: 41
17, 2, {7}: 41
17, {2}, 9: 43
17, 2, {9}: 43
17, {2}, 11: 1238072254113251
17, {2}, 13: 47
17, 2, {13}: 47
17, {2}, 15: 10453
17, 2, {15}: 14431
17, {3}, 1: 919
17, {3}, 2: 53
17, 3, {2}: 53
17, {3}, 4: 15661
17, 3, {4}: 202312198150541083680042289968086175246392709668351015606480937190197719141555582914093984895921012851445550526307546901116396443873602008692570890526616368088185273150378794958020557689359714900361309302539772871024478810285911676119560774228749311819445083611973024639995726017975216038442723332032205483020153342941185706273838443645335316482628378969548962919113661222993675604814390665842420911527121940927081962502185843741384735861252053394208248911431061837614993199706806589642337729249897643547266180540450077659378716054747677348448847073865940465584890988386180646046096142019029751447198046438996788148066525863948926986329748608837564706756460589811903109379913331680366765910754291118931715014909287015368864757215181388931953550080792379895988355601407167699413638955421000867341962147355447891731848892228022595327764226835424944122126463004299092971566585126472463347029693202245076617052610545287249829830436323088053040740425159393510325367055688958021542160943392588303575764092913624724403124665858417454172809264055857172288650133078210277438852308448011492659742540728194755339738118552604075349989025970187600141495195618093003594891737256114574520645691866237754189839522103102401063052231296238777096133716514735976159018372250704239420907521162127191197581548074707774582416270546628549687572701286582548353511119649632719976299664503918217812466555481567
17, {3}, 5: 44826081690965873408843
17, {3}, 7: 12954737608689137415155107
17, {3}, 8: 59
17, 3, {8}: 59
17, {3}, 10: 61
17, 3, {10}: 61
17, {3}, 11: 929
17, {3}, 13: 2059346319065197758094572502134042410628879165166500729310408311859100403174401197154162422913971035803320636990925926115809373804593425373
17, {3}, 14: 15671
17, 3, {14}: 19037
17, {3}, 16: 67
17, 3, {16}: 67
17, {4}, 1: 354961
17, {4}, 3: 71
17, 4, {3}: 71
17, {4}, 5: 73
17, 4, {5}: 73
17, {4}, 7: 1231
17, {4}, 9: 6034397
17, 4, {9}: 0
17, {4}, 11: 79
17, 4, {11}: 79
17, {4}, 13: 1237
17, {4}, 15: 83
17, 4, {15}: 83
17, {5}, 1: 1531
17, 5, {1}: 0
17, 5, {2}: 1481
17, 5, {3}: 1499
17, {5}, 4: 89
17, 5, {4}: 89
17, {5}, 6: 7542991
17, 5, {6}: 1553
17, {5}, 7: 258512581839480238807
17, 5, {7}: 1571
17, 5, {8}: 2192870942958148263240732846802917159482980658678987091029
17, 5, {9}: 1607
17, {5}, 11: 443711
17, 5, {11}: 11465965308803
17, {5}, 12: 97
17, 5, {12}: 97
17, {5}, 13: 1543
17, 5, {13}: 978696116876517899
17, 5, {14}: 1697
17, {5}, 16: 101
17, 5, {16}: 101
17, {6}, 1: 103
17, 6, {1}: 103
17, {6}, 5: 107
17, 6, {5}: 107
17, {6}, 7: 109
17, 6, {7}: 109
17, {6}, 11: 113
17, 6, {11}: 113
17, {6}, 13: 31327
17, 6, {13}: 33469
17, {7}, 1: 2143
17, 7, {1}: 0
17, {7}, 2: 881997331441
17, 7, {2}: 595087
17, 7, {3}: 600307
17, 7, {4}: 95906589091783507930843563755990266422092688356040675515311603708093961802306894007437655217514330623235207811600088376616459742526084918320106897318128806495781475426768767165514390697631437051279309375222665888996529269923522513150777949332557515521324424371314404893207152062522816208660683728587344752060207752176117852541701300415836483279179797153360991805611346793920572915966693512831758798610318187165078100052657265407064700184425873728839407610370409154466626422741733541020144469899847293106180159054565336330268644842349049592211776064686938080891658404158572628900407902382793468029662463581407704575520153279074959618731284192450532813079007371781177428028956685990102815330642281362082111577053562596575982200756771882498844269795552481099163314315787859968104394148733642046151581817378168289655730735285607627344702459668007180874848122337772142808275765179310807830012018795922800290688494743524724295147812117289642950083881206753277901109889820750317055789894070812884761502938943605687373479513650470502061201712723954528850575691371252234690260912366780751178462203664771090103741876657991281157114168815982359929638882464261198071565661897533385653118572545290080553260068584635774174378873241967998873902779616049492290819388713047447836299281834501905197895513178008430148838039753653476431746091649918729698629758813644179453577167011361866263064663482484971061912954834835200477371046714024946618399066857015486252167195960074052560082946015310807355599115365154898206253970363795413248288426431043732650612453678289003108049110516734699599633783987864173701209615158542221041250911073433615334565034714121771220193160128195806230453409786749742845939608758929141872445533757921260002927508261714697148904081140693515371012423699033127330493467490048678198295121841737113578761601362608451647399392411505661604089075312833905939420635095935065291556854483070663161964127222184661472915798063325075795310241827209806620530513237119860789944360183319440790216620845715705884866599976338449612477998170052792542193549653671825352068259451901384834348260380145635795902195520839603052665037992217129182586156768151478332476293754270819051535328015556313203975988535607477203174867043926355262515269619989893528810931352894985781863647407409963294078899410144835689753338000750769507629898619664022834082535232488316695619138887782288109959951826676589454631927024639295210111806877585535436438777573680792913296936989667348185106551317537036603215416045634454099384405880405696970720084684693247251343705527039783036991738239849876055320891925638823902862780492307005254189057301401817013458964613400020256906005181311284022352648807314852064365215297536521853957702230484874643069416504295038457867649364475841900123618490276713195796005557064624890834770622094022000187190171714522228753474852925074066174542928946709438848843012663738859945859221737289291171096569517311534969894552062670173368644363541663192337912778284843268514815921286383827432399361150667678322987438015028340370193694146594687267967014425498222196335366572333217884787193471118715216854187874189418504437464870873131183472181208042966524528356295092674006998162392804478553820447794088341938629744582995866448719357944154114949377566255938868506935999098594262466142856156343589113789200203891385805977216641210094514355114576835097945155598461070241035469794842774507531812654666380339725829035457927631544810648512697742475840941044313592470433686717439064577986319891517211466466148642266977218462335259214786775338256474834203607752993286655331560107205540478606754476315314952772003753859459618476201332195124702019538036820429400127713096356869469174745389203770244409167126770249782096482566537299311052679852899076507988153936077495361427149560460699617095838241560999866892201345400745108331228104164320135716700859602712441769263379666713760107563730203796881758124746645149986594048960910932208225161109728474522151908665166819659957697315780018637157866623837740131680236973888742969842123536647664781356394353599098937132120578421177881953953489697403221141910722457903659248285030041952003816703834491628113647737078152275019652593835963898544432294235176334662225559536463552759541484261645409937761720728736185109624513914201497900199535439579294128406730426260799860203048262445526683425847539709798431034743757191922233526369259328361286021444496038217776974673610871233162722242391857718914109493890692932421845551321497560276396591188063845546634526087874534643700005680620069287168146621038580084350836816579510989510482628397457066953288858121565062507610370561970649072966964965050553424650069839765408206450820404370870725339068554206739587431061242394089506855550109461263470386002891976348462610586433717527777170636075792769147581795163223229658353536650742052756349958237927267250696531810315269126535665858820819115046643587993989863564977869489984571848082751410762948010913445945171202057182212644951369452552196946306607657770384335079326676617600668092435117966493471605287426013406047304987100849679543614747519097587090112318580430667236086364389644686441511407518969127778864009437715768678996806782644813856700895391643481170544085132112509834422205668045202598249661802160961456796770868561640815537805733764286450554023223550825639105798599931452027972612278042719071929656044691267278139961271773129646031075755648907778549016456423452580456170323882015457149188203298018516617745770028717129184349218542367290857962130401275546781641761313111685312570453659481775353751399202949472683653290239318343407935512908574575452351884938053808608726963500511123291053659212626100184745914732686892224327277645008695274244753548102901753214745905371689972868668564767732813722495864293080736509385391464611105049792335724282919607500689731003157257818126550802560310539203000456435191729973185997942574243325084441612690999504386884537127033864111546842429244000200728653230480154878647994887779522578581780048433620098136434153347196499548233867493344400591770455652526102250621509851962443353006687527
17, {7}, 5: 378893998333780600151918874693847240901089686373872230108378808042086874393030756381396925278352076800011918029919353266890458763951613729721003300506667989912895216810823140369059431198254813404980456907218177386238322978141949043394150307
17, 7, {5}: 2113
17, {7}, 6: 3051893879
17, 7, {6}: 2131
17, {7}, 8: 127
17, 7, {8}: 127
17, 7, {9}: 0
17, 7, {10}: 2203
17, {7}, 11: 2153
17, 7, {11}: 2221
17, {7}, 12: 131
17, 7, {12}: 131
17, {7}, 13: 41702102071970378195387359655683833039903949723250346925868452559148327207890893
17, 7, {13}: 652507
17, 7, {15}: 2293
17, 7, {16}: 2311
17, {8}, 1: 137
17, 8, {1}: 137
17, {8}, 3: 139
17, 8, {3}: 139
17, {8}, 5: 205169333
17, {8}, 7: 41759
17, 8, {7}: 41453
17, {8}, 9: 41761
17, 8, {9}: 3513524887
17, {8}, 11: 2459
17, {8}, 13: 149
17, 8, {13}: 149
17, {8}, 15: 151
17, 8, {15}: 151
17, {9}, 1: 1869203091411069355531855107713874939536131449989395150006339082674324728843097134694876907498265745273934952429109315615950710196111138285659470330945631318449307425659030790402840617341437874222598461881879133913175043742346246602676865994997400473054183576876916093555691941736178502386560694611375263595063904161089255385478117171010347061697561996379095781
17, {9}, 2: 0
17, 9, {2}: 26119610345027069111
17, {9}, 4: 157
17, 9, {4}: 157
17, {9}, 5: 318432595751153950765929260897357340270548161611474263261032439
17, {9}, 7: 798667
17, {9}, 8: 0
17, 9, {8}: 1126584826721
17, {9}, 10: 163
17, 9, {10}: 163
17, {9}, 11: 1484977798350500254413793712430566937554737931611808526186904667195181046087793905731792589996736627655426994137073129024366057707897203460483507678035133303365716593172893884937437026459910380639297977860505840938309245704441111978939720110627977640039737901597151324407412762783691381348217201857297710952740180161422146895372735157410289658892002936744181900705726584432634813253077897930046233672820880239776911825001489964358127541526621813068048456677805001704305921861565527867539831639250429173215267040916636092040704485096023324307405256389881176818589340226230788760221017153318401853554456294869299859474316434325775618344335121116055307007022846189770019398196291526515166484820726469725893644511218051827243865879539556725725237671
17, {9}, 13: 2767
17, {9}, 14: 167
17, 9, {14}: 167
17, {9}, 16: 15055951007560998946554855935799468825319902264000113359851960469680860934897071208938883683385877468609842462917887776927689487889
17, 9, {16}: 9173262075447700370870228797372658996167179560206469728679179339455142654588221994000682651957214660337554289179686628006320784508450532253801769
17, {10}, 1: 3061
17, {10}, 3: 173
17, 10, {3}: 173
17, {10}, 7: 3067
17, {10}, 9: 179
17, 10, {9}: 179
17, {10}, 11: 181
17, 10, {11}: 181
17, {10}, 13: 12479730563037202376599416073
17, {11}, 1: 568727680046856525361
17, 11, {1}: 1862679706313372777
17, 11, {2}: 929171
17, 11, {3}: 78041286371
17, {11}, 4: 191
17, 11, {4}: 191
17, {11}, 5: 3371
17, 11, {5}: 166550976124720415238207768204709708607521852138914978863354513192214473215336519626508334154498477187662381682580264286031705945239873635193075307951288200708854083160986114702227981399575722978160695129072904545350759971963253886646257214591
17, {11}, 6: 193
17, 11, {6}: 193
17, {11}, 7: 3373
17, 11, {7}: 955271
17, 11, {8}: 3323
17, 11, {9}: 965711
17, {11}, 10: 197
17, 11, {10}: 197
17, {11}, 12: 199
17, 11, {12}: 199
17, {11}, 13: 9394230696635382053176380469368734655867242678435691492562299088334773
17, 11, {13}: 3413
17, 11, {14}: 991811
17, 11, {15}: 3449
17, 11, {16}: 3467
17, {12}, 1: 3673
17, 12, {1}: 59263
17, {12}, 5: 3677
17, 12, {5}: 1378486138632359758323050626992747918650304829615263354995388341922232251095972398990848507942018139080087311
17, {12}, 7: 211
17, 12, {7}: 211
17, {12}, 11: 62639
17, 12, {11}: 434824684403093
17, {12}, 13: 88940907373
17, 12, {13}: 18191917
17, {13}, 1: 3768651696722334407412704432886748501027917638216745188121713071850567124051602405518201798458848401
17, 13, {1}: 635636818875898469533
17, {13}, 2: 223
17, 13, {2}: 223
17, 13, {3}: 1101433
17, 13, {4}: 1106653
17, {13}, 5: 27845915749943
17, 13, {5}: 3847
17, {13}, 6: 227
17, 13, {6}: 227
17, {13}, 7: 228154556301155739164141873957905004400241046704207
17, 13, {7}: 93736740613
17, {13}, 8: 229
17, 13, {8}: 229
17, 13, {9}: 3919
17, 13, {10}: 1137973
17, {13}, 11: 3989
17, 13, {11}: 1143193
17, {13}, 12: 233
17, 13, {12}: 233
17, {13}, 14: 160688404748616050182618301672566324918805941
17, 13, {14}: 5693449087
17, 13, {15}: 4027
17, 13, {16}: 1169293
17, {14}, 1: 239
17, 14, {1}: 239
17, {14}, 3: 241
17, 14, {3}: 241
17, {14}, 5: 4289
17, {14}, 9: 21120367
17, 14, {9}: 39353705070153506531713748224825668451495187254508108657151444786252940850766831171765242341910761541585017932281107
17, {14}, 11: 103764391931
17, {14}, 13: 251
17, 14, {13}: 251
17, {14}, 15: 533707265356695216704103124038332368542873946283142922292003783538887189897335797029414828786775114395339696927112171531204140647898504891296804256353517276617145553123762661125816798731498754318189024538693935214938297276615812780293
17, 14, {15}: 73387
17, {15}, 1: 4591
17, {15}, 2: 257
17, 15, {2}: 257
17, {15}, 4: 546208347402889
17, 15, {4}: 74923
17, {15}, 7: 4597
17, {15}, 8: 263
17, 15, {8}: 263
17, {15}, 11: 111176134211
17, {15}, 13: 4603
17, {15}, 14: 269
17, 15, {14}: 269
17, {15}, 16: 271
17, 15, {16}: 271
17, {16}, 1: 34271896307617
17, {16}, 3: 74443609190419550764562450397778200846849192983001551466849044370008879517232307105675227070196683723355515193456559323901778769141226118951876996802487398051974943265833071289084569071200666892787
17, 16, {3}: 3110633786280773828357619125469664392231273829727161657043436458261702025678677414669396669249
17, {16}, 5: 277
17, 16, {5}: 277
17, {16}, 7: 4903
17, {16}, 9: 281
17, 16, {9}: 281
17, {16}, 11: 283
17, 16, {11}: 283
17, {16}, 13: 4909
17, {16}, 15: 24137567
17, 16, {15}: 66886068539071498820247358361862720864806052666582265636907882027208271253
[/CODE]

Found by the smallest prime of the form x{0}y in base b: (written in base 10)

[CODE]
17, 1, 2: 19
17, 1, 4: 293
17, 1, 6: 23
17, 1, 8: 410338681
17, 1, 10: 582622237229771
17, 1, 12: 29
17, 1, 14: 31
17, 1, 16: 83537
17, 2, 1: 13555929465559461990942712143872578804076607708197374744547
17, 2, 3: 37
17, 2, 5: 167047
17, 2, 7: 41
17, 2, 9: 43
17, 2, 11: 196201332019845680883945379695949
17, 2, 13: 47
17, 2, 15: 593
17, 3, 2: 53
17, 3, 4: 6047981701351
17, 3, 8: 59
17, 3, 10: 61
17, 3, 14: 881
17, 3, 16: 67
17, 4, 1: 96550277
17, 4, 3: 71
17, 4, 5: 73
17, 4, 7: 1163
17, 4, 9: 19661
17, 4, 11: 79
17, 4, 13: 2200203825088579408776819567241541693183008366424296889532537802676605742013242605885738467254947449275279256209
17, 4, 15: 83
17, 5, 2: 1447
17, 5, 4: 89
17, 5, 6: 1451
17, 5, 8: 1453
17, 5, 12: 97
17, 5, 14: 1459
17, 5, 16: 101
17, 6, 1: 103
17, 6, 5: 107
17, 6, 7: 109
17, 6, 11: 113
17, 6, 13: 1747
17, 7, 2: 2872370713
17, 7, 4: 2027
17, 7, 6: 2029
17, 7, 8: 127
17, 7, 10: 1384392410141922893881019214409802183915866577
17, 7, 12: 131
17, 7, 16: 2039
17, 8, 1: 137
17, 8, 3: 139
17, 8, 5: 76989925372723444096295785745849952492079199962951999650667399693
17, 8, 7: 93068397320465404630656753909902667826276751
17, 8, 9: 39313
17, 8, 11: 668179
17, 8, 13: 149
17, 8, 15: 151
17, 9, 2: 751691
17, 9, 4: 157
17, 9, 8: 2609
17, 9, 10: 163
17, 9, 14: 167
17, 9, 16: 2617
17, 10, 1: 30813241584205620768578082879858566794519475523316935428523219222549715730421553051581012062309582291563417357962735106623882511119803053532925182926249776721567747363668587303886289198263003405664727396667247087240804081542494465987591751010474069802881434021018098552503636187479633948985542054586040960634216385275240883080107491030029424981368312950391012273198193145074691992034596972340401058372496572072805178747749043809661175062774693744253875524750296244557254164154416700682064649608501244980115298005252325285889222942497166632190100307172425095654640555255216809069690029471120246262607934101091950191883243664485672079467291700033284161575098430234471556894428035387297153958363118712933585316020997979453186050455259481808205798419933449883468946598312334293337309542762515158871429229577473038775624210435828319336665689500445775721699294960847296671163818571043532412360970879096440103385837923437659493729312640502280289332144186542909375183018988331924064567217852850883038801075973312283384453560413672274330910190835800312461177887946949267924016085740244494693777167147182081956756446500290483708727538440256482902518426838263191879732255320705694276850481260357774187393051990398145941391579734666612247292602622909196256365550090854044975724104967579044465191427546422562330455332142861211297760555314684220699373702898852004167554484798100653125570311064985359245562181761217076160929488301267141304344823042013963475979223346004598840045846455460225949484184828624913029196546544529865367330320626191913535613530529262162961396970046656462697047971179581727354850607452126759187784236625512474620330422245358784789547477050410772212033297916811
17, 10, 3: 173
17, 10, 7: 2897
17, 10, 9: 179
17, 10, 11: 181
17, 10, 13: 2903
17, 11, 2: 3181
17, 11, 4: 191
17, 11, 6: 193
17, 11, 8: 3187
17, 11, 10: 197
17, 11, 12: 199
17, 11, 14: 22175932904953
17, 11, 16: 54059
17, 12, 1: 3469
17, 12, 5: 1002257
17, 12, 7: 211
17, 12, 11: 58967
17, 12, 13: 17038297
17, 13, 2: 223
17, 13, 4: 3761
17, 13, 6: 227
17, 13, 8: 229
17, 13, 10: 3767
17, 13, 12: 233
17, 13, 14: 5334402763
17, 13, 16: 106515054435591952235202013374298845053
17, 14, 1: 239
17, 14, 3: 241
17, 14, 5: 4051
17, 14, 9: 68791
17, 14, 11: 4057
17, 14, 13: 251
17, 14, 15: 138664092460683133
17, 15, 2: 257
17, 15, 4: 4339
17, 15, 8: 263
17, 15, 14: 269
17, 15, 16: 271
17, 16, 1: 1336337
17, 16, 3: 1336339
17, 16, 5: 277
17, 16, 7: 1336343
17, 16, 9: 281
17, 16, 11: 283
17, 16, 13: 4637
17, 16, 15: 4639
[/CODE]

Found by CRUS generalized Sierpinski/Riesel problem base b: (written in base b)

[CODE]
F7(0^186767)1
97(0^166047)1
57(0^51310)1
53(0^4867)1
[/CODE]

sweety439 2021-09-05 20:27

Found by the smallest prime of the form xy{0}z or x{0}yz which no possible prime subsequence (i.e. no possible prime of the form x{0}y, x{0}z, y{0}z): (written in base 10)

[CODE]
17: 11{0}1: 307
17: 1{0}11: 307
17: 1{0}13: 4933
17: 11{0}5: 311
17: 1{0}15: 311
17: 11{0}7: 313
17: 1{0}17: 313
17: 1{0}19: 1419883
17: 11{0}B: 317
17: 1{0}1B: 317
17: 11{0}D: 71081873576577829004543318111499865716103849328323588573540479792752701567
17: 1{0}1D: 4943
17: 1{0}1F: 0
17: 13{0}1: 198091560658118741
17: 1{0}31: 14063084452067724991061
17: 13{0}3: 5783
17: 1{0}33: 4967
17: 1{0}35: 4969
17: 13{0}7: 347
17: 1{0}37: 347
17: 13{0}9: 349
17: 1{0}39: 349
17: 13{0}B: 5791
17: 1{0}3B: 1419919
17: 13{0}D: 353
17: 1{0}3D: 353
17: 1{0}3F: 410338739
17: 15{0}1: 6359
17: 1{0}51: 4999
17: 15{0}3: 6361
17: 1{0}53: 83609
17: 15{0}5: 379
17: 1{0}55: 379
17: 15{0}7: 0
17: 1{0}57: 103257064210705262567352386053079314585953296995896382999756683132532869411895922908989006972530155003188434343319672948902496772622784175883224955759851948757249557799022126004597063933467084017079022283555685909778396894148559539227141497282350107280187042887783246626841561924612635569101211255317264657756145394890919230981531485747616532526706894914665610797974036930896717009140991099334338492482687211101378491982463571217949369591901487391667725736473097704869553685241068337315322860538695432604791831537112343571971769614998557985339306932281462253458349016975318839588817429133797571713571461079379113485566267732699526967693711606732673618273293956217957939561648764484339589657288339036527182344974105613620657525858072226774186666886713406898208077219230684461025162216908651093044568201310295450686823793873116823069869
17: 15{0}9: 383
17: 1{0}59: 383
17: 1{0}5B: 5009
17: 15{0}D: 0
17: 1{0}5D: 5011
17: 15{0}F: 389
17: 1{0}5F: 389
17: 17{0}1: 409
17: 1{0}71: 409
17: 1{0}73: 3362095853201812742282475234995233875224247499
17: 17{0}5: 117917
17: 1{0}75: 4773695331839566234818968439734627784374274207965213
17: 17{0}7: 2004511
17: 1{0}77: 5039
17: 1{0}79: 139288917338851014461418017489467720561
17: 17{0}B: 419
17: 1{0}7B: 419
17: 17{0}D: 421
17: 1{0}7D: 421
17: 1{0}7F: 410338807
17: 19{0}1: 443
17: 1{0}91: 443
17: 19{0}3: 7517
17: 1{0}93: 410338829
17: 19{0}5: 2171551
17: 1{0}95: 34271896307791
17: 19{0}7: 449
17: 1{0}97: 449
17: 19{0}9: 7523
17: 1{0}99: 24137731
17: 19{0}B: 2171557
17: 1{0}9B: 5077
17: 1{0}9D: 582622237229927
17: 19{0}F: 457
17: 1{0}9F: 457
17: 1B{0}1: 8093
17: 1{0}B1: 5101
17: 1B{0}3: 479
17: 1{0}B3: 479
17: 1B{0}5: 39756001
17: 1{0}B5: 36926505171389432251064150202562814007472190016867877501650856866068714188255953768088437087198019533295018612411027616169487104837532697335951669148723663600469236692245142217080369441
17: 1{0}B7: 5107
17: 1B{0}9: 8101
17: 1{0}B9: 83717
17: 1B{0}B: 487
17: 1{0}BB: 487
17: 1B{0}D: 67745860443544174916725123307335114827143631083190735496519618006200558084850571839519663430081617004350734224061438452895725404102658308093247938061809837512466345790905697257026568511471957013675880159815573776628578414983502181344946027965193870223263794583119067741099320362183757067919509085157022107703313465537254620408715810477636550531626549946146031833469366570313563183200221469649796037979025570918536497582497494578162524171298602170690304025740841072301298009111814882587792341405970528724088953452364350440719062655660326732882987507173190967336910882132801589291653231520518309174212814775976372806629949863647029684420593217346752797860129930941296085265468966863644593073977021777772170320559662551718071750570525758544826881250997859964026509622155820857278533183837591557975023077660238870335888384712629399912742987669901684102718463459487718369297044104406767570829425829060499979775646992615659041220244273275020005497803525273042500150977306933678373041007498781477514624092388564826581196982467398536693538259774915345482799959533153196081448869753956773729168510405992946692355178779639795875198875224527357156935519943470861903347828889845442772932870387685208836544671288017490302229193397019475672333966567016865767135624917056982357106647819150281446379495951761366537128412444160391334429343678930705939453825277451921991792143079246812322809117658177177765379477042858824955943356003051987806013200430646134249048990165055214720294669034389435409426226057635514431221081611248628655212153211454138821500471403655169505068702500675177980231290879082280099770547422395572502157230264514314901298523979875516756287521064008070376162889713005401350342083112102360020626006751094040988956345109537896018537664858064173924637293018614181830081760055872090257311096048306384057037975618317336362784087125753702550705024128260870613052589015634090615667974106859621078145456715024592968866327819335692144546754746217298418634855847305022921013725678622988449334003967485750187004181264478534798466287115460917041437006892624849105746334172002792031827536433434552181497649698460712498304135450331007699960828082342823655374658860933477885278618050564655003581576200757524468690584518821514191544607511929325339937808831345585082632204238366391202767708898229512936647466373111326484480039123655885625018127234717992524868050281138815448361455401
17: 1{0}BD: 5113
17: 1B{0}F: 491
17: 1{0}BF: 491
17: 1D{0}1: 147391
17: 1{0}D1: 2015993900671
17: 1{0}D3: 410338897
17: 1{0}D5: 28351092476867700887730107366063267
17: 1D{0}7: 8677
17: 1{0}D7: 14063084452067724991237
17: 1{0}D9: 2862423051509816023
17: 1D{0}B: 521
17: 1{0}DB: 521
17: 1D{0}D: 523
17: 1{0}DD: 523
17: 1{0}DF: 1420093
17: 1F{0}1: 157217
17: 1{0}F1: 83777
17: 1F{0}3: 547
17: 1{0}F3: 547
17: 1F{0}5: 75773171032334951867011401514270439915557
17: 1{0}F5: 528244191897564154934050393570530948631222822294909349211666370710303517436667795444973593088437096234383886074919719698316973578068291172490891450223728364291053020348665704329605029703032760236174765342650636295195316985703869124677738552176423987060750650356623052974426941490900929321334684120278689634806660631844362188141051100906075178024087696526687900937167187775247063313539192481410716315845117034618387215424748571822390226877839425703801545296924750698132592917
17: 1F{0}7: 0
17: 1{0}F7: 582622237230023
17: 1F{0}9: 9257
17: 1{0}F9: 1420121
17: 1F{0}B: 772402219
17: 1{0}FB: 5179
17: 1F{0}D: 557
17: 1{0}FD: 557
17: 1F{0}F: 157231
17: 1{0}FF: 83791
17: 23{0}6: 181787
17: 23{0}C: 641
17: 25{0}A: 673
17: 27{0}E: 11863
17: 29{0}6: 12433
17: 29{0}C: 743
17: 2F{0}6: 839
17: 2F{0}A: 4806932634486219181656661802550491
17: 2F{0}C: 14173
17: 31{0}1: 1255153589
17: 31{0}3: 887
17: 3{0}13: 887
17: 31{0}5: 73832569
17: 3{0}15: 355763629513
17: 31{0}7: 4343099
17: 31{0}9: 211340033145673771164883261
17: 3{0}19: 4259597
17: 31{0}B: 255487
17: 3{0}1B: 14767
17: 31{0}F: 21337611011
17: 3{0}1F: 14771
17: 32{0}6: 907
17: 32{0}C: 15329
17: 33{0}1: 919
17: 3{0}31: 919
17: 33{0}5: 31461600810407099
17: 3{0}35: 250619
17: 33{0}7: 2112107890693109202596989096303314769567949965605605231165546218438757586787222997747178301
17: 3{0}37: 14797
17: 33{0}B: 929
17: 3{0}3B: 929
17: 33{0}D: 15619
17: 3{0}3D: 2914937104725971647558906028740867769819422475923
17: 34{0}6: 941
17: 34{0}C: 947
17: 35{0}1: 953
17: 3{0}51: 953
17: 35{0}3: 16187
17: 3{0}53: 14827
17: 35{0}5: 16189
17: 3{0}57: 14831
17: 35{0}9: 16193
17: 3{0}59: 72412801
17: 35{0}B: 106584268343391390044551812945957037040007325709135489458323630579489551516212444575779718136917869515065431107
17: 35{0}D: 22978965701
17: 3{0}5D: 1231016117
17: 35{0}F: 967
17: 3{0}5F: 967
17: 37{0}1: 16763
17: 37{0}3: 284957
17: 3{0}73: 5547972117396809516573440924088504787922044378857996800825375654057364531608506481098397349246362298087961231118993836544197347696126641115913572733347998541
17: 37{0}5: 991
17: 3{0}75: 991
17: 37{0}7: 2664566743290736991606506832595921654466901605695793657404382802351615428590976517449650056592206755708348681972818689
17: 37{0}9: 4844227
17: 3{0}79: 14867
17: 37{0}B: 997
17: 3{0}7B: 997
17: 37{0}D: 15102094906093617476986560844830752913008537486738989396364821341906625677938706777856022481075628386307668965917531058345270128635845064555873729874069222749888544747218042717511226946479949976157059089666482952075814752591644693999762923053297743456305088122790485501859785712489994482146407117312191709280484700946538483477390425948610550594071272275834552842917188065880488971829172697707000039077782031400149656468836129898169335122467789137658603675153501018563380255333096921257885512758525856228150127141952519470245544548199779772519356079531045931137865415828111097959750347865582774456155924082506218673780472652243299070392563362518269406650517545703667247409825548018774616109532685929511402933174072149529986276938066083082506524352277415804102144023174960726197980671674790942932248257783745721425186859717905471035223308200048185347803934225807873639185871797882352626530663710127332498573070907395383905966903634486453983553547904895182979370011319683348898807992072648220369892728537406806376621388670987251846218725363554158243913574879031674972705439406055438144089427324267916089522196919362917650200730250617215844277588619557131251725630161422706525727520528437078353432409609333739535612206072959435788669499366288679805374496984397860584520673362072463276289508843514759014892207065599097648199659723026985339564715095839838433500828231
17: 37{0}F: 284969
17: 3{0}7F: 3326698118342350923918335147625284899308051502949638573208309523836662741504100093137274082768015280767745354509205192824137
17: 38{0}6: 1009
17: 38{0}C: 4927751
17: 39{0}1: 1021
17: 3{0}91: 1021
17: 3{0}95: 14897
17: 39{0}7: 294787
17: 3{0}97: 72412867
17: 39{0}B: 1031
17: 3{0}9B: 1031
17: 39{0}D: 1033
17: 3{0}9D: 1033
17: 3A{0}6: 299699
17: 3A{0}C: 1049
17: 3B{0}1: 177470229193608579167
17: 3{0}B1: 250751
17: 3B{0}3: 17921
17: 3{0}B3: 14929
17: 3B{0}5: 17923
17: 3B{0}7: 1061
17: 3{0}B7: 1061
17: 3B{0}9: 1063
17: 3{0}B9: 1063
17: 3B{0}B: 17929
17: 3B{0}D: 66802353160605310243398378264555925584132921208159754425974664266324699031229640652131269792286317951416353026439412094967624186556585484849362901502642563611498263866789099
17: 3{0}BD: 14939
17: 3B{0}F: 1069
17: 3{0}BF: 1069
17: 3D{0}1: 10776180899801659457
17: 3D{0}3: 1091
17: 3{0}D3: 1091
17: 3D{0}5: 1093
17: 3{0}D5: 1093
17: 3D{0}7: 18503
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17: BF{0}9: 286811123
17: BF{0}B: 992437
17: B{0}FB: 25445202703884930170114288224989158223507372694843530790472898237205746664426794115271729250206159926331135789335582267662188652123200422252774464216813085420612927319162754641272652347330191067844077616232230625710643687265349121741024410436920927916611413755176155714106156976892917719545177716624643807151697098742845163703862131758299142966698780190086571972420843184301593104625301576578810250747677870490419132588072298036445664052442606723891932239892381703875907827228971175947252618668909430796321022201333352491905257768410550874445052740633273690295269889880265430859317814521972031731107306031826626728343010201438454441343514558731187509025122342026506258197
17: BF{0}D: 58391
17: B{0}FD: 54311
17: BF{0}F: 3449
17: B{0}FF: 3449
17: C{0}23: 1002289
17: C{0}29: 3511
17: C{0}2F: 3517
17: C{0}32: 59009
17: C{0}34: 59011
17: C{0}38: 3527
17: C{0}3A: 3529
17: C{0}3E: 3533
17: C{0}3G: 59023
17: C{0}43: 3539
17: C{0}49: 17038361
17: C{0}4F: 90192970090916585998935437091543612474998225158236216165237183470959611288229482030747251179803833584274419570309857804606929627122001296159458585341885602117979002708516528924275359
17: C5{0}A: 348551666333255852090328967029833867
17: C7{0}E: 299589841
17: C{0}83: 3607
17: C{0}89: 3613
17: C{0}8F: 59107
17: C{0}92: 3623
17: C{0}94: 59113
17: C{0}98: 6991466846757293
17: C{0}9A: 3631
17: C{0}9E: 59123
17: C{0}9G: 3637
17: C{0}A3: 98321588709777186678648012345506626361
17: C{0}A9: 289651007
17: CA{0}F: 61861
17: C{0}AF: 59141
17: C{0}E3: 3709
17: C{0}E9: 9926883142636041170371
17: C{0}EF: 59209
17: C{0}F2: 83709089549
17: C{0}F4: 3727
17: C{0}F8: 59219
17: CF{0}A: 3733
17: C{0}FA: 3733
17: C{0}FE: 1423054518233
17: C{0}FG: 3739
17: C{0}G3: 1002527
17: C{0}G9: 28414939137125606950129275567851414968613
17: C{0}GF: 59243
17: D1{0}1: 7608360980294527
17: D{0}11: 43707246091623565649672178054938040377915215919
17: D{0}13: 114765423924637301941867999296200327467925167817679991074231087625410120274777671637980461622707901225495301352410635755286433490835973170690405664980932313013624278466313523272026046995652138904837782443175281
17: D1{0}5: 3779
17: D{0}15: 3779
17: D1{0}7: 18541669
17: D{0}17: 313788421
17: D1{0}B: 1090697
17: D{0}1B: 1085801
17: D1{0}D: 64171
17: D{0}1D: 6265591437387761896188353727899932091
17: D{0}1F: 63901
17: D3{0}1: 1100513
17: D3{0}3: 18708707
17: D{0}33: 1085827
17: D3{0}5: 18708709
17: D{0}35: 18458197
17: D{0}37: 7574089083986951
17: D3{0}9: 1562569666793
17: D{0}39: 63929
17: D3{0}B: 64747
17: D{0}3B: 18458203
17: D3{0}D: 3821
17: D{0}3D: 3821
17: D3{0}F: 3823
17: D{0}3F: 3823
17: D5{0}1: 10997429363900712276707
17: D{0}51: 103494894795404556444206444926698977031395728463686168989122211064009500977127571487347
17: D5{0}3: 18875749
17: D{0}53: 7574089083986981
17: D5{0}5: 3847
17: D{0}55: 3847
17: D5{0}7: 5455090601
17: D{0}57: 18458233
17: D5{0}9: 3851
17: D{0}59: 3851
17: D5{0}B: 3853
17: D{0}5B: 3853
17: D5{0}D: 65327
17: D{0}5D: 10998431815262684602062432053335281153637354392528616594832217955625845823076518030629242866726545306498226461049468577944661363499914612249378425606252110439039903961577365184833281562289165346323268458117710464347693979166008440567597409649919177875046417807469366030175035112698798580734280626092922611080834882261315076734981180507900205003624926131147362808972187014276195477842657835589530283143068291559869091466689438505852077849621295147169051866418617734537754078224227693051142003068886635764252741737474848857490648400993328660650451029958930665997731573522644769775499288743333041330972590559979013089425887718700381210116385298922182320141997455575976663418594201999610643375011892637992200496635968891978925292989634739160644796162761351669723458978023501729202588409212014240086181978176624509474953242878433453953925626352528473445518573659493207961146847
17: D5{0}F: 1110353
17: D{0}5F: 1085873
17: D7{0}1: 3877
17: D{0}71: 3877
17: D{0}73: 259578395711173809433267854391
17: D7{0}5: 3881
17: D{0}75: 3881
17: D7{0}7: 65899
17: D{0}77: 18458267
17: D{0}79: 63997
17: D7{0}B: 19042799
17: D7{0}D: 3889
17: D{0}7D: 3889
17: D{0}7F: 0
17: D9{0}1: 3911
17: D{0}91: 3911
17: D9{0}3: 27275211594313
17: D{0}95: 1810755925405063187998434227363080365787
17: D9{0}7: 3917
17: D{0}97: 3917
17: D9{0}9: 3919
17: D{0}99: 3919
17: D9{0}B: 3234509423975576747932081
17: D{0}9B: 64033
17: D9{0}D: 3923
17: D{0}9D: 3923
17: D{0}9F: 64037
17: DB{0}1: 67049
17: D{0}B1: 5334402937
17: DB{0}3: 3947
17: D{0}B3: 3947
17: DB{0}5: 95198572141
17: D{0}B5: 1541642394653
17: DB{0}7: 186477991154601808456634889317094882580637458943065704921878364790901999
17: D{0}B7: 64063
17: DB{0}9: 67057
17: D{0}B9: 313788593
17: DB{0}B: 2698983522293496734289045863387177366962025587
17: D{0}BB: 64067
17: DB{0}D: 67061
17: D{0}BD: 3107941663906967223013189
17: DB{0}F: 329406839
17: D{0}BF: 432901829517908638949760040958498867818784428857429063779704439
17: DD{0}1: 26198234025581497132799500240918011205211884680606832492907278944958566234027008435643334079872669607694654283
17: D{0}D1: 64091
17: D{0}D3: 208645076635146541586862529156359778603621737755944370276634633735671536279673489578862383305680413
17: DD{0}5: 67631
17: D{0}D5: 7150662431537883078524663593535010502844777190878964890980747858698968661543038469128650018578579210144657582863
17: DD{0}7: 96019249489
17: D{0}D7: 18458369
17: D{0}D9: 259578395711173809433267854499
17: DD{0}B: 3989
17: D{0}DB: 3989
17: D{0}DF: 996008028344507597478088093200970407624542705262405925671717858893822860550021755909210320506354083339868733322549588915030693715809385975680710922057
17: DF{0}1: 4013
17: D{0}F1: 4013
17: DF{0}3: 68207
17: D{0}F3: 1086031
17: DF{0}5: 68209
17: DF{0}7: 4019
17: D{0}F7: 4019
17: DF{0}9: 4021
17: D{0}F9: 4021
17: DF{0}B: 19710967
17: D{0}FB: 5519728037387365233350388697171115520181178003002275953858616347325030826717144661809808621903839257164131861080601399376640850115886221082616456713756819407322108781731149761358716846406867703109903481801385373526093324728867001271618131287816947118866201785961593625328875887930171823125000314259608036422530687331216007971330473704093006084627666743229144591082897897812692002130212512481343741082345310990601510152408832768052982555169720966454026254978803152273482995137504336749786015236877650051401952632116033171485780769872446927139687
17: DF{0}D: 1615453376102110361265244453483387073832266432748438446740262751460576539761648244268574817919673281482272494566161489317209374503323989321080731871807249290285388107121717645570421808883145704274961365958081596269092194608751744829618174695161639017442100861577530449170608058608544667127826536360400540488202685269091327144349248008358096638722586074545745817997127952428425501278789807643735346493519252313498547293568593818838133477070874988460016127091523625491041316652889482507135217500092994469158557058112072186031624745750431130845229148642755248307725459566272913725963097273354705830270332919178810048263218955754952479181449837601290462968437118648777597783675451475150170513275394527021617887869117854070186188704898342777251272521336796177327224410741985853846774462704832305113026761247424769466733762393639720648783146713986473851900867931845116786734785686711338529896727440317484635730724585166426158335015400116920665168752895260683398498767417961754967220055228924276051709566248871048572029761517064386581634340971160106760919623653905752494374569726268153172867511185986858946442610542553134548144952237282461806293590262552710308930674205535614931723559077600133516197862729002538265697210227685500412904672234853046533215163794817637434261009523758552837564987113510609714038252055746994008786408872506435478425800280654582883215625989132005573634411204115691145573674458676425987327627111120732662856329458588300580253738863700768576365784403303570496686498891436599339506354699137882360945384294956122256579040917819184904335952713933848023628806877455680914072700609108850958171473075792610154735913302719860839731025483103286097718443346357813625099347342902702271855848201202723749948833776789408559357175470805112123249926555466931624832860838007974242548253977149403817778058600121336226558701753906894699992906446231601001069171840088407744479637331643468505491166766600460401453058850910549306407599290218033514780075484467739816112153327725434791637890800042328578237366179719627503446695816361937309899229108046768651640704357430712550188610687432830621806642946315889791878754429158897566213410829369265631305198428417696984244815330301560799708977500302280826358279539802647357505567265089758884610025752407450922100195552252677437779661759887620246931785486878331985285269506865556819719658431460942123451833841581110683810686733492472600610230437478900537587335844427425674411081213355676681522386992588430355348724406896976066998074136041182254129509656971372377393060615936706494569258464240335508324724448567051268258748917390234085765770730067263268415467399424896709885531565507838926920051719944203788420753431306413276161088936396947458344203007553727618361
17: D{0}FD: 508470418129822585810386263924872074155247213942090148243557422957478678300627758716913561
17: DF{0}F: 4027
17: D{0}FF: 4027
17: E{0}27: 337926007
17: E3{0}6: 296347534367821302635809466782971037082760544496011547037767247152688950205327493554021676997764068285433873777069567918453979319586603853991209999000334990192124875403755299883915382055975718372155511134404139897305638761725581932892488288671056247
17: E3{0}C: 69661
17: E{0}47: 1169369
17: E5{0}A: 70237
17: E{0}67: 68891
17: E{0}72: 68903
17: E{0}74: 1169417
17: E{0}76: 1169419
17: E{0}78: 68909
17: E{0}7A: 23347958510361636025189500183816751
17: E{0}7C: 4177
17: E{0}7G: 68917
17: E{0}87: 28223914606429
17: E9{0}6: 71389
17: E9{0}C: 4211
17: E{0}A7: 97660604351
17: E{0}C7: 68993
17: EF{0}6: 21130819
17: EF{0}A: 73127
17: EF{0}C: 103815684281
17: E{0}G7: 69061
17: F1{0}1: 836292091916227193884933339554840428416401773403713206576249295218837202517597165012584667463937
17: F1{0}3: 21381379
17: F1{0}5: 4357
17: F{0}15: 4357
17: F1{0}7: 21381383
17: F1{0}9: 8773605454754057
17: F{0}19: 73721
17: F1{0}B: 4363
17: F{0}1B: 4363
17: F1{0}D: 21381389
17: F1{0}F: 73999
17: F{0}1F: 73727
17: F2{0}6: 74279
17: F2{0}A: 105457038971
17: F2{0}C: 4245120070182590042728286813189617095447018932342669
17: F3{0}1: 366323107
17: F{0}31: 21297907
17: F3{0}5: 4391
17: F{0}35: 4391
17: F3{0}7: 520126426315849
17: F{0}37: 1252873
17: F3{0}B: 4397
17: F{0}3B: 4397
17: F3{0}D: 150316537205278351
17: F{0}3D: 362063599
17: F4{0}6: 4409
17: F4{0}A: 74861
17: F4{0}C: 150899159442508111
17: F5{0}1: 4421
17: F{0}51: 4421
17: F5{0}3: 4423
17: F{0}53: 4423
17: F5{0}7: 1277387
17: F{0}57: 8739333558446507
17: F5{0}9: 75149
17: F{0}59: 21297949
17: F5{0}B: 21715471
17: F{0}5D: 1252913
17: F6{0}A: 4447
17: F{0}6A: 4447
17: F7{0}1: 0
17: F7{0}3: 4457
17: F{0}73: 4457
17: F7{0}5: 254571173812734857220144459843369118564229562891691
17: F{0}75: 73819
17: F7{0}7: 46184281963746569621030034681067943178662090816762303760121077122282811708167396433683693721230900699184378482202376461713960138945804563839098771065800709886739459418771498485856681561544711633673870354309517013262511992218052699540818189
17: F7{0}9: 4463
17: F{0}79: 4463
17: F7{0}B: 1287217
17: F7{0}D: 75731
17: F7{0}F: 21882517
17: F{0}7F: 30239908506869
17: F8{0}6: 6348180653
17: F8{0}A: 4481
17: F8{0}C: 4483
17: F9{0}1: 127239703036182241584132721858890928009
17: F{0}91: 73849
17: F9{0}5: 4493
17: F{0}95: 4493
17: F9{0}7: 76303
17: F{0}9B: 73859
17: F9{0}D: 153812270628656917
17: F{0}9D: 21298021
17: FA{0}6: 1848575721871
17: F{0}A6: 1252991
17: FA{0}C: 4517
17: F{0}AC: 4517
17: FB{0}1: 4523
17: F{0}B1: 4523
17: FB{0}3: 6420593357
17: FB{0}5: 24349684185825681118448288226082259409141596127431803942915332254845888110317472716779987318876963225381725830222999754780108401355493550245881843432922087997737806587097445846468010657122822665335174988823299364375983408177542311048568795088764938787646744735634827733955076346404353567370169832329956420969342096557053568889498776018576080674825198745665276974644234706955301494160329164639582713233168586009859335718545710221961089043461741071
17: F{0}B7: 86559411185226035684093549866049
17: FB{0}9: 76883
17: F{0}B9: 1253011
17: FB{0}B: 10404087017117908294274057400308920901945827608353536879445098038976102186766691062976841237
17: FB{0}D: 98251443979244338860839715792752292963016369034889921324186431097616252543291084792106542158380968269640017036084483884070349326562311784195875103747727346652591287192807153312720178190658456638224358400020069502519329147487661273615764063794600398865031334552699697285245978237030934891823581289264409436910875674882192628533707065499611886523781345147908254457997788946857901742314474580483467617865803662177476054508010019081771114078937139156776613850125582609850223011040975887067271608757412107377813856410353825374800175568439460618674578632109093371830126653553962681891898754807084962442871262475778080247873225445373690925154379525562513214303401737700795783103688480861419686057629638418123171423761264886165562323561374199718147576593195336913084299442310787862210797238112198695929242601938712093560608800442220455728850611202372556468106908404565400577805824420128933926860578788734393413761617592348363125537611400024761452965572420164809850527261492961346880703094457842500843810701824678352916018992047199882398785572056999805128053129608463330955238272421797182762359517823660111188532766857651544266763259597903022340299966950805113519127017133367170410791444282401279347882163847950103971046972649733986073134562183812437008592625455555279882095501160435721490990147495868262188672412275937019143114467825548036950316328624571278127068724749862787135718017431891746966199443819749572074721740797770479736202736354386724448820315335440820713701561654292424851260871681436123846054989254542730307368625370595749674451793407627838570474082034056105194353110561127543154606363290486618891073666065235141911287600698116375086073584302167209777817357199621341264349786045583129060965557894746335913340889674621364272845060412671569820359039754449706349956862197965031343580171116938229746387029473455624545906429559482230184743576679790466200604456413508692084231231574022336357102440244711522159910447536642303952566449037500861040956612680331417049816333596112313332090273711092215656179701128690743101917869841228167592389182306380330231349919543447000477749928762968878570469383931168178937098702356629180328503630397351455128548978853318960325705736201723694938849764991478221213016323713131129257108230189579200652898287269467107806506337545123227735329352723438900764149694531443791889101503104485272880379081378308049270751534382732404051451991205604464014225000000645146564876422906426556336931034957194681024159735963233939410414623844065115043758573492580143069242917683920750858457103456118486792787132938172051357924963748209759172429896334625283156305257826485503406522544930267761042303903859006456421673136573785607454268995965773709156197025968992745377680809087242371113499641940499384290843586974692930315227440933374791701553576031938009451071330568795224276804391375424233125713093880712863417160462068917782391
17: FB{0}F: 1306873
17: F{0}BF: 73897
17: FC{0}A: 4549
17: F{0}CA: 4549
17: FD{0}1: 22383629
17: FD{0}3: 380521679
17: F{0}D3: 6155080319
17: FD{0}5: 4561
17: F{0}D5: 4561
17: FD{0}7: 0
17: FD{0}9: 22383637
17: FD{0}B: 4567
17: F{0}DB: 4567
17: FD{0}D: 1219799581314837067929646554325645081413240272602297293691688199133133620115093712014372933812666347762969534621672421427924701497549183579435588579021708156035590276688594658271547454175815650916221938454929107862937470289997417491057623032371335460608280935152596523866704920657744610968633795755708712581059162825409315531712905443169556905885316485157952849267004847881474887336655048329663172379721138580063170553919376582339806823532037272748206778403475238851231571461323795797072704736121558893190390072435736611289141788146683109941874725403834167414015402645637218761867947378771245462428200835479435980679264149631101034817638210945319221160912238250649967712981354469725647357192242618176494609761400348105634145662991640258211606594362645812722414866310213896973839882840519792016203246590421737600984367140731972560738646214625511673799179875236878589211161968238647114634356246492407610246200574475235982914914257937620528547087551645448535277555826530801023741137694596544493451746433776926533311450572435833239651962167048285826880073389221377838744148165406700720674759906027191559115567013721470919945617414923167932413200800275360784915044972487597647131868688916865332614766646796507404379440901672080411616495894238247349649875449065031857702227662087845894584365127569206986027371637800614789335276183069009800341391989640579000994438012380141801877495721819833817159586472517051027850525696731605873597592115837725152751972468331064753468169170156804848262124166435011920437352594243117563441147758411552476554468116794565610818281557385129150139601453300336171134373643183167346802286944989583006217487791599620645017239131074031330353936918594369138389114185462560032614924736296561439488659783135266165164606142252279942291557832704569691559998044403509608509320812455041748967703157144299957937835532539692159548688296282434639943806071770258792757096929576188407946275757892529225336113856969682779566741436702285804126183536256451396372132841917594272671598735601785240522717016277327315171393110962300818792858846431146873464227904257653862499344557036624754382354766956588774295921421172109491676241381128071593673411523342785514361851328724737590812570677519431714726775517425268932402807393388974143365113890791490788823270111758353060469398926393667658929327431665390956444066233443431537288144788189284387671125612907343415391617368401897482772431168492707578921443816264870657682328164084391987382887931801955451860662610176054107047373293605863972234301574251622912440194597815005580608127982377511100536898758333188398838564855121573986414099924978795815847945357764102755226467018023016904629244578070203656216379037651969015798684929397209857517861414931223482412875856454287018434542087630370416610516961220740735392252304291973814378092763683025624601
17: FD{0}F: 1316699
17: F{0}DF: 21298091
17: FE{0}6: 77747
17: FE{0}A: 4583
17: FE{0}C: 22467161
17: FF{0}1: 4591
17: F{0}F1: 4591
17: FF{0}7: 4597
17: F{0}F7: 4597
17: FF{0}B: 78041
17: F{0}FB: 73961
17: FF{0}D: 4603
17: F{0}FD: 4603
17: FG{0}6: 22634197
17: FG{0}A: 571961122735202782903853471534900926841794718508083962199240831790562247996749386040420834804611195259068938674235881304819860071499732379798058145217214758134439722774254794523366692857088136115587644686054597897175594906427623384921
17: FG{0}C: 45630390997597651771
17: G3{0}6: 79481
17: G3{0}C: 1351087
17: G5{0}A: 1932284811167
17: G7{0}E: 1370741
17: G9{0}6: 4783
17: G9{0}C: 4789
17: GF{0}6: 1410037
17: GF{0}A: 4889
17: GF{0}C: 1410043
[/CODE]

Other known primes:

[CODE]
1(0^9019)1F
9(7^309)3
76(9^122)
[/CODE]

sweety439 2021-09-05 20:31

Known minimal primes (start with b+1) in base b=18:

Small ones: (written in base b)

[CODE]
11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, ...
[/CODE]

Found by original minimal primes search: (written in base b)

[CODE]
GG0000000000000000000000000000001
[/CODE]

Found by smallest generalized near-repdigit primes (i.e. of the form x{y} or {x}y) base b: (written in base 10) (the numbers whose repeating digit (i.e. y for x{y}, or x for {x}y) is 1 are not minimal primes (start with b+1), but still post here)

[CODE]
18, {1}, 1: 19
18, 1, {1}: 19
18, {1}, 5: 23
18, 1, {5}: 23
18, {1}, 7: 349
18, 1, {7}: 457
18, {1}, 11: 29
18, 1, {11}: 29
18, {1}, 13: 31
18, 1, {13}: 31
18, {1}, 17: 359
18, 1, {17}: 647
18, {2}, 1: 37
18, 2, {1}: 37
18, {2}, 5: 41
18, 2, {5}: 41
18, {2}, 7: 43
18, 2, {7}: 43
18, {2}, 11: 47
18, 2, {11}: 47
18, {2}, 13: 72025897
18, 2, {13}: 3198298525119427
18, {2}, 17: 53
18, 2, {17}: 53
18, {3}, 1: 18523
18, 3, {1}: 991
18, {3}, 5: 59
18, 3, {5}: 59
18, {3}, 7: 61
18, 3, {7}: 61
18, {3}, 11: 108038837
18, 3, {11}: 1181
18, {3}, 13: 67
18, 3, {13}: 67
18, {3}, 17: 71
18, 3, {17}: 71
18, {4}, 1: 73
18, 4, {1}: 73
18, {4}, 5: 1373
18, 4, {5}: 47321007179
18, {4}, 7: 79
18, 4, {7}: 79
18, {4}, 11: 83
18, 4, {11}: 83
18, {4}, 13: 1381
18, 4, {13}: 1543
18, {4}, 17: 89
18, 4, {17}: 89
18, {5}, 1: 30871
18, 5, {1}: 34130064295121260303
18, {5}, 7: 97
18, 5, {7}: 97
18, {5}, 11: 101
18, 5, {11}: 101
18, {5}, 13: 103
18, 5, {13}: 103
18, {5}, 17: 107
18, 5, {17}: 107
18, {6}, 1: 109
18, 6, {1}: 109
18, {6}, 5: 113
18, 6, {5}: 113
18, {6}, 7: 3889397851
18, 6, {7}: 412073923449193
18, {6}, 11: 2063
18, 6, {11}: 2153
18, {6}, 13: 37057
18, 6, {13}: 39451
18, {6}, 17: 2069
18, 6, {17}: 2267
18, {7}, 1: 127
18, 7, {1}: 127
18, {7}, 5: 131
18, 7, {5}: 131
18, {7}, 11: 137
18, 7, {11}: 137
18, {7}, 13: 139
18, 7, {13}: 139
18, {7}, 17: 2411
18, 7, {17}: 2591
18, {8}, 1: 49393
18, 8, {1}: 845983
18, {8}, 5: 149
18, 8, {5}: 149
18, {8}, 7: 151
18, 8, {7}: 151
18, {8}, 11: 889211
18, 8, {11}: 2801
18, {8}, 13: 157
18, 8, {13}: 157
18, {8}, 17: 2753
18, 8, {17}: 578415690713087
18, {9}, 1: 163
18, 9, {1}: 163
18, {9}, 5: 167
18, 9, {5}: 167
18, {9}, 7: 1000357
18, 9, {7}: 3049
18, {9}, 11: 173
18, 9, {11}: 173
18, {9}, 13: 55579
18, 9, {13}: 3163
18, {9}, 17: 179
18, 9, {17}: 179
18, {10}, 1: 181
18, 10, {1}: 181
18, {10}, 7: 3968612127339681427
18, 10, {7}: 3373
18, {10}, 11: 191
18, 10, {11}: 191
18, {10}, 13: 193
18, 10, {13}: 193
18, {10}, 17: 197
18, 10, {17}: 197
18, {11}, 1: 199
18, 11, {1}: 199
18, {11}, 5: 3767
18, 11, {5}: 3659
18, {11}, 7: 3769
18, 11, {7}: 3697
18, {11}, 13: 211
18, 11, {13}: 211
18, {11}, 17: 3779
18, 11, {17}: 132239526911
18, {12}, 1: 140018322601
18, 12, {1}: 3907
18, {12}, 5: 74093
18, 12, {5}: 71699
18, {12}, 7: 223
18, 12, {7}: 223
18, {12}, 11: 227
18, 12, {11}: 227
18, {12}, 13: 229
18, 12, {13}: 229
18, {12}, 17: 233
18, 12, {17}: 233
18, {13}, 1: 4447
18, 13, {1}: 4231
18, {13}, 5: 239
18, 13, {5}: 239
18, {13}, 7: 241
18, 13, {7}: 241
18, {13}, 11: 4457
18, 13, {11}: 4421
18, {13}, 17: 251
18, 13, {17}: 251
18, {14}, 1: 4789
18, 14, {1}: 26565103
18, {14}, 5: 257
18, 14, {5}: 257
18, {14}, 11: 263
18, 14, {11}: 263
18, {14}, 13: 4801
18, 14, {13}: 4783
18, {14}, 17: 269
18, 14, {17}: 269
18, {15}, 1: 271
18, 15, {1}: 271
18, {15}, 7: 277
18, 15, {7}: 277
18, {15}, 11: 281
18, 15, {11}: 281
18, {15}, 13: 283
18, 15, {13}: 283
18, {15}, 17: 5147
18, 15, {17}: 30233087
18, {16}, 1: 32011489
18, 16, {1}: 30344239
18, {16}, 5: 293
18, 16, {5}: 293
18, {16}, 7: 5479
18, 16, {7}: 95713
18, {16}, 11: 5483
18, 16, {11}: 5393
18, {16}, 13: 32011501
18, 16, {13}: 5431
18, {16}, 17: 98801
18, 16, {17}: 5507
18, {17}, 1: 307
18, 17, {1}: 307
18, {17}, 5: 311
18, 17, {5}: 311
18, {17}, 7: 313
18, 17, {7}: 313
18, {17}, 11: 317
18, 17, {11}: 317
18, {17}, 13: 5827
18, 17, {13}: 9770144707511081415118442597789015238720654947319882836100223544506052645981243442054558121499672250712069138857313219
[/CODE]

Found by the smallest prime of the form x{0}y in base b: (written in base 10)

[CODE]
18, 1, 1: 19
18, 1, 5: 23
18, 1, 7: 331
18, 1, 11: 29
18, 1, 13: 31
18, 1, 17: 5849
18, 2, 1: 37
18, 2, 5: 41
18, 2, 7: 43
18, 2, 11: 47
18, 2, 13: 661
18, 2, 17: 53
18, 3, 1: 17497
18, 3, 5: 59
18, 3, 7: 61
18, 3, 11: 983
18, 3, 13: 67
18, 3, 17: 71
18, 4, 1: 73
18, 4, 5: 1301
18, 4, 7: 79
18, 4, 11: 83
18, 4, 17: 89
18, 5, 1: 1621
18, 5, 7: 97
18, 5, 11: 101
18, 5, 13: 103
18, 5, 17: 107
18, 6, 1: 109
18, 6, 5: 113
18, 6, 7: 1951
18, 6, 13: 11337421
18, 6, 17: 629873
18, 7, 1: 127
18, 7, 5: 131
18, 7, 11: 137
18, 7, 13: 139
18, 7, 17: 40841
18, 8, 1: 2593
18, 8, 5: 149
18, 8, 7: 151
18, 8, 11: 16965591777169791368456755523227278360431637956235834935249793817382094395433673839081451555581626681225386838273435065340853159062269727290105590614749873345235150074200837919455814998922832689807722347561507360730864684612838991122461180927466240001961713930606333114920563153920496212361659607159524282838957737069091620601882623971333680788347168728743159252297352852013067
18, 8, 13: 157
18, 8, 17: 2609
18, 9, 1: 163
18, 9, 5: 167
18, 9, 7: 306110023
18, 9, 11: 173
18, 9, 13: 52501
18, 9, 17: 179
18, 10, 1: 181
18, 10, 11: 191
18, 10, 13: 193
18, 10, 17: 197
18, 11, 1: 199
18, 11, 5: 64157
18, 11, 7: 3571
18, 11, 13: 211
18, 11, 17: 3581
18, 12, 1: 3889
18, 12, 7: 223
18, 12, 11: 227
18, 12, 13: 229
18, 12, 17: 233
18, 13, 1: 46416073946113
18, 13, 5: 239
18, 13, 7: 241
18, 13, 11: 4872573778567053323
18, 13, 17: 251
18, 14, 1: 81649
18, 14, 5: 257
18, 14, 11: 263
18, 14, 13: 4549
18, 14, 17: 269
18, 15, 1: 271
18, 15, 7: 277
18, 15, 11: 281
18, 15, 13: 283
18, 15, 17: 4877
18, 16, 5: 293
18, 16, 7: 93319
18, 16, 11: 93323
18, 16, 13: 5197
18, 16, 17: 93329
18, 17, 1: 307
18, 17, 5: 311
18, 17, 7: 313
18, 17, 11: 317
18, 17, 13: 5521
[/CODE]

Found by CRUS generalized Sierpinski/Riesel problem base b: (written in base b)

(none)

Found by the smallest prime of the form xy{0}z or x{0}yz which no possible prime subsequence (i.e. no possible prime of the form x{0}y, x{0}z, y{0}z): (written in base 10)

[CODE]
18: 4{0}2D: 419953
18: 4{0}3D: 7558339
18: 44{0}D: 1381
18: 4{0}4D: 1381
18: 4{0}6D: 7558393
18: 4{0}8D: 1453
18: 4{0}9D: 1471
18: 4{0}AD: 1489
18: 4{0}CD: 23557
18: 4D{0}D: 1543
18: 4{0}DD: 1543
18: 4{0}ED: 23593
18: 4{0}FD: 1579
18: 4{0}GD: 1597
18: 5{0}25: 29201
18: 5{0}35: 524939
18: 5{0}45: 1697
18: 5{0}65: 1733
18: 5{0}85: 525029
18: 5{0}95: 1787
18: 5C{0}5: 33053
18: 5{0}C5: 525101
18: 5{0}E5: 1877
18: 5{0}G5: 1913
18: 6{0}2B: 629903
18: 6{0}3B: 629921
18: 6{0}4B: 2027
18: 66{0}B: 2063
18: 6{0}6B: 2063
18: 6{0}8B: 2099
18: 6{0}9B: 630029
18: 6{0}AB: 1128032249017374725365639668832841738254486483902838030139765293247
18: 6B{0}B: 2153
18: 6{0}BB: 2153
18: 6{0}CB: 3673320419
18: 6{0}EB: 2207
18: 6{0}FB: 842910363109321117699883607005331737
18: 6{0}GB: 2243
18: 7{0}27: 2311
18: 7{0}47: 2347
18: 7{0}67: 2383
18: 7{0}87: 13227127
18: 7{0}97: 2437
18: 7A{0}7: 44071
18: 7{0}A7: 41011
18: 7{0}C7: 41047
18: 7{0}F7: 735109
18: 7{0}G7: 1388515032871
18: A{0}27: 58363
18: A{0}35: 3299
18: A{0}37: 3301
18: A{0}45: 1049837
18: A{0}47: 3319
18: A{0}57: 58417
18: A{0}65: 11568313814261873
18: A{0}75: 3371
18: A7{0}7: 3373
18: A{0}77: 3373
18: A{0}85: 3389
18: A{0}87: 3391
18: A{0}95: 3407
18: A{0}97: 1983592903849
18: AA{0}7: 19945447
18: A{0}A7: 18895867
18: AC{0}5: 3461
18: A{0}C5: 3461
18: A{0}C7: 3463
18: A{0}E5: 13382588450523947024397379830017
18: AE{0}7: 3499
18: A{0}E7: 3499
18: A{0}F7: 3517
18: A{0}G5: 3533
18: B{0}2B: 4122947043402891311
18: B{0}3B: 64217
18: B{0}4B: 1154819
18: B6{0}B: 66107
18: B{0}6B: 64271
18: B{0}8B: 3719
18: B{0}9B: 4769554523766734719495226171326637
18: B{0}AB: 1154927
18: B{0}EB: 20785511
18: B{0}FB: 64433
18: B{0}GB: 3863
18: C{0}25: 3929
18: C{0}35: 3947
18: C{0}45: 70061
18: C5{0}5: 1288877
18: C{0}55: 70079
18: C{0}65: 4001
18: C{0}85: 132239527061
18: C{0}95: 22674983
18: CA{0}5: 4073
18: C{0}A5: 4073
18: CC{0}5: 73877
18: C{0}C5: 0
18: C{0}E5: 70241
18: CF{0}5: 141422827397
18: C{0}F5: 158977933792616233827672418718890662663562426562825809340354517616087501660156563830771135501287406841875017088863453099475288561977198545621995225363
18: C{0}G5: 22675109
18: D{0}2D: 4261
18: D{0}3D: 75883
18: D4{0}D: 4297
18: D{0}4D: 4297
18: D{0}6D: 75937
18: D{0}9D: 75991
18: D{0}AD: 24564577
18: D{0}CD: 4441
18: D{0}ED: 76081
18: D{0}FD: 76099
18: D{0}GD: 4513
18: E{0}27: 2328932811259158381328230697097177961909091820544372409528282741884072046383387351101015789410446161335422421149956871021828688543106068943794449299527044803380369101538653195437536877210639731824418021959947964238599612915195153978540619229078169497877795554070227102121162405101325994322599811138104207587347818917419501362640626129052538301052878891
18: E{0}37: 4597
18: E{0}47: 81727
18: E{0}67: 4651
18: E{0}87: 81799
18: E{0}97: 81817
18: EA{0}7: 4723
18: E{0}A7: 4723
18: E{0}C7: 4759
18: E{0}F7: 4813
18: E{0}G7: 4831
18: F{0}25: 1574681
18: F{0}35: 4919
18: F{0}45: 4937
18: F{0}65: 4973
18: F{0}85: 5009
18: F{0}95: 0
18: FC{0}5: 5081
18: F{0}C5: 5081
18: F{0}G5: 5153
18: G{0}11: 1189563417824350846613100429331
18: G{0}21: 1679653
18: G{0}31: 176319369271
18: G{0}41: 18509302102818889
18: G{0}61: 30233197
18: G{0}81: 30233233
18: G{0}91: 5347
18: G{0}A1: 93493
18: G{0}C1: 93529
18: G{0}E1: 5437
18: G{0}F1: 9795520783
18: GG{0}1: 249069897374447078426903207266791381270529
18: G{0}G1: 93601
18: H{0}2H: 32122709
18: H{0}3H: 2064472028642102280263
18: H{0}4H: 99233
18: H{0}6H: 32122781
18: H{0}8H: 5669
18: H{0}9H: 187339329971
18: H{0}AH: 1784789
18: H{0}CH: 5741
18: H{0}EH: 578208077
18: H{0}FH: 99431
18: H{0}GH: 5813
[/CODE]

sweety439 2021-09-05 20:35

Known unsolved families in bases b=17 and b=18:

b=17:

[CODE]
F1{9}
1{7}
15{0}D
1F{0}7
51{0}D
73{0}B
9D{0}5
B3{0}D
B{0}B3
B{0}DB
[/CODE]

b=18:

[CODE]
C{0}C5
[/CODE]

sweety439 2021-09-08 05:55

Some unsolved families are obvious through [URL="https://en.wikiversity.org/wiki/Quasi-minimal_prime"]data for minimal primes (start with b+1) base b up to certain limit[/URL]

Example 1: 80555551 is minimal prime (start with b+1) base b for b=10, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 80{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 8{5}1 searched to 5 5's (the smallest prime in this family is 8(5^20)1, but this prime is not minimal prime (start with b+1) for base b=10 since (5^11)1 is prime in this base and 20 >= 11)
* 0{5}1 searched to 5 5's (not considered, since this family has [URL="https://en.wikipedia.org/wiki/Leading_zero"]leading zeros[/URL])
* 8{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 0{5} searched to 5 5's (not considered, since this family has leading zeros)
* {5}1 searched to 5 5's (the smallest prime in this family is (5^11)1, and indeed minimal prime (start with b+1) for base b=10)
* {5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 2: 55555025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {5}02 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}05 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* {5}25 searched to 5 5's (the smallest prime in this family is (5^13)25, and indeed minimal prime (start with b+1) for base b=8)
* {5}0 searched to 5 5's (not considered, since this family has [URL="https://en.wikipedia.org/wiki/Trailing_zero"]trailing zeros[/URL])
* {5}2 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}5 searched to 5 5's = {5} searched to 6 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 3: 33333301 is minimal prime (start with b+1) base b for b=7, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {3}0 searched to 6 3's (not considered, since this family has trailing zeros)
* {3}1 searched to 6 3's (the smallest prime in this family is (3^16)1, and indeed minimal prime (start with b+1) for base b=7)
* {3} searched to 6 3's (this family cannot have primes since all such numbers are divisible by 3)

Example 4: 100000000000507 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 1{0}50 searched to 11 0's (not considered, since this family has trailing zeros)
* 1{0}57 searched to 11 0's (the smallest prime in this family is 1(0^25)57, and indeed minimal prime (start with b+1) for base b=9)
* 1{0}07 searched to 11 0's = 1{0}7 searched to 12 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}5 searched to 11 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}0 searched to 11 0's = 1{0} searched to 12 0's (not considered, since this family has trailing zeros)

Example 5: BBBBBB99B is minimal prime (start with b+1) base b for b=12, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {B}99 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}9B searched to 6 B's (this family cannot have primes since such numbers with even length are factored as difference of squares and such numbers with odd length are divisible by 13)
* {B}9 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}B searched to 6 B's = {B} searched to 7 B's (this family cannot have primes since all such numbers are divisible by 11)

Example 6: 500025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 5{0}2 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 2)
* 5{0}5 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 5)
* {0}25 searched to 3 0's (not considered, since this family has leading zeros)
* 5{0} searched to 3 0's (not considered, since this family has trailing zeros)
* {0}2 searched to 3 0's (not considered, since this family has leading zeros)
* {0}5 searched to 3 0's (not considered, since this family has leading zeros)
* {0} searched to 3 0's (not considered, since this family has leading zeros and trailing zeros)

Example 7: 77774444441 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {7}1 searched to 4 7's
* {7}41 searched to 4 7's
* {7}441 searched to 4 7's
* {7}4441 searched to 4 7's
* {7}44441 searched to 4 7's
* {7}444441 searched to 4 7's
* {4}1 searched to 6 4's
* 7{4}1 searched to 6 4's
* 77{4}1 searched to 6 4's
* 777{4}1 searched to 6 4's

Example 8: 88888888833335 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {8}5 searched to 9 8's
* {8}35 searched to 9 8's
* {8}335 searched to 9 8's
* {8}3335 searched to 9 8's
* {3}5 searched to 4 3's
* 8{3}5 searched to 4 3's
* 88{3}5 searched to 4 3's
* 888{3}5 searched to 4 3's
* 8888{3}5 searched to 4 3's
* 88888{3}5 searched to 4 3's
* 888888{3}5 searched to 4 3's
* 8888888{3}5 searched to 4 3's
* 88888888{3}5 searched to 4 3's

Example 9: A44444777 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* A{4} searched to 5 4's
* A{4}7 searched to 5 4's
* A{4}77 searched to 5 4's
* A{7} searched to 3 7's
* A4{7} searched to 3 7's
* A44{7} searched to 3 7's
* A444{7} searched to 3 7's
* A4444{7} searched to 3 7's

Example 10: 96664444 is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* 9{6} searched to 3 6's
* 9{6}4 searched to 3 6's
* 9{6}44 searched to 3 6's
* 9{6}444 searched to 3 6's
* 9{4} searched to 4 4's
* 96{4} searched to 4 4's
* 966{4} searched to 4 4's

Example 11: 88828823 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {8} searched to 5 8's
* {8}2 searched to 5 8's
* {8}3 searched to 5 8's
* 8882{8} searched to 5 8's
* {8}288 searched to 5 8's
* {8}23 searched to 5 8's
* 8882{8}2 searched to 5 8's
* 8882{8}3 searched to 5 8's
* {8}2882 searched to 5 8's
* {8}2883 searched to 5 8's

Example 12: B0BBB05BB is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they [I]possible[/I] contain primes:

* {B} searched to 6 B's
* B0{B} searched to 6 B's
* {B}0BB searched to 6 B's
* {B}5BB searched to 6 B's
* BBBB0{B} searched to 6 B's
* BBBB5{B} searched to 6 B's
* B0{B}0BB searched to 6 B's
* B0{B}5BB searched to 6 B's
* BBBB05{B} searched to 6 B's
* B0BBB0{B} searched to 6 B's
* B0BBB5{B} searched to 6 B's

(for the reference of leading zeros and trailing zeros, see [URL="https://oeis.org/A141709"]https://oeis.org/A141709[/URL] [URL="https://oeis.org/A061816"]https://oeis.org/A061816[/URL] [URL="https://oeis.org/A061906"]https://oeis.org/A061906[/URL] [URL="https://oeis.org/A004151"]https://oeis.org/A004151[/URL] [URL="https://oeis.org/A213321"]https://oeis.org/A213321[/URL] [URL="https://oeis.org/A340164"]https://oeis.org/A340164[/URL] [URL="https://oeis.org/A339996"]https://oeis.org/A339996[/URL] [URL="https://oeis.org/search?q=%22leading+zero%22+%22trailing+zero%22&sort=&language=&go=Search"]https://oeis.org/search?q=%22leading+zero%22+%22trailing+zero%22&sort=&language=&go=Search[/URL])

sweety439 2021-09-10 05:37

2 Attachment(s)
Newest data (minimal primes (start with b+1) up to certain limit) for bases b=17 and b=18

sweety439 2021-09-20 03:10

Newest condensed table for bases 2<=b<=16:

[CODE]
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a*bn+c)/d) form of largest known minimal prime base b
2 1 11 2 3
3 3 111 3 13
4 5 221 3 41
5 22 1(0^93)13 96 5^95+8
6 11 40041 5 5209
7 ≥71 (3^16)1 17 (7^17-5)/2
8 75 (4^220)7 221 (4*8^221+17)/7
9 ≥149 3(0^1158)11 1161 3*9^1160+10
10 77 5(0^28)27 31 5*10^30+27
11 ≥914 55(7^1011) 1013 (607*11^1011-7)/10
12 106 4(0^39)77 42 4*12^41+91
13 ≥2496 8(0^32017)111 32021 8*13^32020+183
14 ≥605 4(D^19698) 19699 5*14^19698-1
15 ≥1171 (7^155)97 157 (15^157+59)/2
16 ≥2050 D(B^32234) 32235 (206*16^32234-11)/15
[/CODE]

Known values or lower bounds of the largest minimal prime base b for 2<=b<=36:

[CODE]
base (b) largest known minimal prime written in base b largest known minimal prime written in decimal length of largest known minimal prime written in base b
2 11 3 2
3 111 13 3
4 221 41 3
5 1(0^93)13 5^95+8 96
6 40041 5209 5
7 (3^16)1 (7^17-5)/2 17
8 (4^220)7 (4*8^221+17)/7 221
9 3(0^1158)11 3*9^1160+10 1161
10 5(0^28)27 5*10^30+27 31
11 55(7^1011) (607*11^1011-7)/10 1013
12 4(0^39)77 4*12^41+91 42
13 8(0^32017)111 8*13^32020+183 32021
14 4(D^19698) 5*14^19698-1 19699
15 (7^155)97 (15^157+59)/2 157
16 D(B^32234) (206*16^32234-11)/15 32235
17 F7(0^186767)1 262*17^186768+1 186770
18 8(0^298)B 8*18^299+11 300
19 FG(6^110984) (904*19^110984-1)/3 110986
20 C(D^2449) (241*20^2449-13)/19 2450
21 C(F^479147)0K (51*21^479149-1243)/4 479150
22 K(0^760)EC1 22^763*20+7041 764
23 9(E^800873) (106*23^800873-7)/11 800874
24 2(0^313)7 2*24^314+7 315
25 9(6^136965)M (37*25^136966+63)/4 136967
26 (M^8772)P (22*26^8773+53)/25 8773
27 A(0^109003)PM 10*27^109005+697 109006
28 O4(O^94535)9 (6092*28^94536-143)/9 94538
29 O(0^174236)FPL 24*29^174239+13361 174240
30 O(T^34205) 25*30^34205-1 34206
31 IE(L^29787) (5727*31^29787-7)/10 29789
32 S(U^9748)L (898*32^9749-309)/31 9750
33 N7(0^610411)1 766*33^610412+1 610414
34 US(0^9374)R 1048*34^9375+27 9377
35 1B(0^56061)1 46*35^56062+1 56064
36 (P^81993)SZ (5*36^81995+821)/7 81995
[/CODE]

jwaltos 2021-09-20 03:39

Sweety, could you please explain what it is you're doing? More specifically, what is your end objective, goal or result?
You have a lot of "stuff" but how does it support whatever it is you are doing? Second, by posting, what is it that you hope others, like myself, will gain by reading it all? Cheers.

sweety439 2021-09-20 15:52

1 Attachment(s)
[QUOTE=jwaltos;588209]Sweety, could you please explain what it is you're doing? More specifically, what is your end objective, goal or result?
You have a lot of "stuff" but how does it support whatever it is you are doing? Second, by posting, what is it that you hope others, like myself, will gain by reading it all? Cheers.[/QUOTE]

I have a project: To find and proof all minimal primes (start with b+1) in bases 2<=b<=36, this text file is the current data, only bases 2, 3, 4, 5, 6, 8, 10, 12 were solved.

You can see [URL="https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub"]my article[/URL].

sweety439 2021-09-20 18:54

1 Attachment(s)
upload the text file for the proof for bases 2, 3, 4, 5, 6, 8, 10, 12

jwaltos 2021-09-21 00:45

Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it.

sweety439 2021-09-22 20:00

[QUOTE=jwaltos;588277]Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it.[/QUOTE]

I have a conjecture in my article: every x{y}z (i.e. xyyy...yyyz) simple family in every base b (the algebraic form is (a*b^n+c)/gcd(a+c,b-1) with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) which can be proven to contain no primes > base (by covering congruence, by algebra factorization, or by the combine of them) contains a prime > base, in fact, I conjectured that every x{y}z (i.e. xyyy...yyyz) simple family in every base b (the algebraic form is (a*b^n+c)/gcd(a+c,b-1) with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) which can be proven to contain no primes > base or only contain finite primes > base (by covering congruence, by algebra factorization, or by the combine of them) contains infinitely many primes > base.

sweety439 2021-09-22 20:04

[QUOTE=jwaltos;588277]Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it.[/QUOTE]

You can help me to completely solve bases 7, 9, and 11 through 36, I think that my data for base 7 is complete, but I cannot prove this.

jwaltos 2021-09-22 23:28

You were kind enough to respond so I'll try to help out. I can't predict when I'll have something but when I do I'll send it via PM rather than posting within the forum.

sweety439 2021-10-02 17:52

1 Attachment(s)
PARI/GP program of this problem attached (not completely done, continue updating ....)

Note: in this program, vector [a,b,[c],d,e] means simple family ab{c}de = abccc...cccde, and vector [a,b,[c,d],e,f] means nonsimple family ab{c,d}ef, and all numbers in these vectors are the digit value of the base b digits, i.e. 10 for A, 11 for B, 12 for C, ...

sweety439 2021-10-05 12:07

[QUOTE=sweety439;569244]Some known unsolved families for bases b<=64 not in the [URL="https://github.com/curtisbright/mepn-data/tree/master/data"]list for bases 2 to 30[/URL] or [URL="https://github.com/RaymondDevillers/primes"]list for bases 28 to 50[/URL]:

Base 11: 5{7} (found by me)
Base 13: 9{5} (found by me)
Base 13: A{3}A (found by me)
Base 17: 15{0}D (found by me)
Base 17: 1F{0}7 (found by me)
Base 18: C{0}C5 (found by me)
Base 25: F{2} (found by extended generalized Riesel conjecture base 25 with k > CK)
Base 31: 2{F} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 31[/URL])
Base 31: 3{5} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 31[/URL])
Base 32: 8{0}V (see [URL="https://oeis.org/A247952"]https://oeis.org/A247952[/URL])
Base 32: S{V} (found by [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024"]CRUS generalized Riesel conjecture base 1024[/URL])
Base 37: 2K{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S37"]CRUS generalized Sierpinski conjecture base 37[/URL])
Base 37: {I}J (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])
Base 38: 1{0}V (see [URL="https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31"]https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31[/URL])
Base 43: 2{7} (found by [URL="https://docs.google.com/document/d/e/2PACX-1vReofbA92gRRhzqjKA3TKOqsukineM59WpM56LuRnbhB7bBFSYL6w-aTJ2IpJPWpiyCmPLOSE6gqDrR/pub"]extended generalized Riesel conjecture base 43[/URL])
Base 43: 3b{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S43"]CRUS generalized Sierpinski conjecture base 43[/URL])
Base 53: 19{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S53"]CRUS generalized Sierpinski conjecture base 53[/URL])
Base 53: 4{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S53"]CRUS generalized Sierpinski conjecture base 53[/URL])
Base 55: a{0}1 (found by [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S55"]CRUS generalized Sierpinski conjecture base 55[/URL])
Base 55: {R}S (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])
Base 60: Z{x} (see [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base60-reserve.htm"]CRUS generalized Riesel conjecture base 60[/URL])
Base 62: 1{0}1 (see [URL="http://jeppesn.dk/generalized-fermat.html"]http://jeppesn.dk/generalized-fermat.html[/URL])
Base 63: {V}W (see [URL="http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt"]http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt[/URL])[/QUOTE]

No, base 32 family 8{0}V has covering set {3, 5, 41}, thus can be ruled out as only contain composite numbers.

sweety439 2021-10-06 23:52

New minimal prime (start with b+1) in base b is found for b=908: 8(0^243438)1, see post [URL="https://mersenneforum.org/showpost.php?p=589662&postcount=992"]https://mersenneforum.org/showpost.php?p=589662&postcount=992[/URL]

File [URL="https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml"]https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml[/URL] updated.

sweety439 2021-10-16 13:38

Conjecture: If sequence (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) does not have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), then the sum of the reciprocals of the positive integers n such that (a*b^n+c)/gcd(a+c,b-1) is prime is converge (i.e. not infinity) and transcendental number. (of course, this conjecture will imply that [URL="https://mersenneforum.org/showpost.php?p=529838&postcount=675"]there are infinitely many such n[/URL])

For the examples of (a,b,c) triples (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) such that (a*b^n+c)/gcd(a+c,b-1) have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), see post [URL="https://mersenneforum.org/showpost.php?p=529847&postcount=678"]https://mersenneforum.org/showpost.php?p=529847&postcount=678[/URL]

sweety439 2021-10-16 13:45

Another conjecture (seems to already be proven, but I am not sure that): If all but finitely many primes p divide (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) for some n>=1, then a=1 and c=-1, i.e. (a*b^n+c)/gcd(a+c,b-1) is generalized repunit number (b^n-1)/(b-1)

The [URL="https://stdkmd.net/nrr/#factortables"]factor tables[/URL] have many examples for the special case that b=10, e.g. [URL="https://stdkmd.net/nrr/2/22221.htm"]{2}1 in base 10[/URL] is (a,b,c) = (2,10,-11), the section [URL="https://stdkmd.net/nrr/2/22221.htm#prime_period"]Prime factors that appear periodically[/URL] lists the primes that divide (a*b^n+c)/gcd(a+c,b-1) = (2*10^n-11)/9 for some n, and we note that the primes 2, 5, 11, 31, 37, 41, 43, 53, 71, 73, 79, 83, 101, 103, 107, 127, 137, 157, 173, 191, 199, 227, 239, 241, 251, 271, 281, 283, 307, 311, 317, 331, 347, 349, 353, 397, 409, 449, 523, 547, 563, 569, 599, 601, 613, 617, 631, 641, 643, 653, 661, 673, 691, 719, 733, 739, 751, 757, 761, 769, 773, 787, 797, 809, 827, 829, 839, 853, 859, 907, 911, 967, 991, 997, ..., divides no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (2*10^n-11)/9, and this sequence of primes seems to be infinite, another example is [URL="https://stdkmd.net/nrr/1/13333.htm"]1{3} in base 10[/URL] is (a,b,c) = (4,10,-1), the section [URL="https://stdkmd.net/nrr/2/13333.htm#prime_period"]Prime factors that appear periodically[/URL] lists the primes that divide (a*b^n+c)/gcd(a+c,b-1) = (4*10^n-1)/3 for some n, and we note that the primes 2, 3, 5, 11, 37, 41, 53, 73, 79, 101, 103, 137, 139, 173, 211, 239, 241, 271, 277, 281, 317, 331, 349, 353, 397, 421, 449, 463, 521, 547, 607, 613, 617, 661, 673, 733, 751, 757, 773, 797, 829, 853, 859, 907, 967, ..., divides no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (4*10^n-1)/3, and this sequence of primes seems to be infinite.

Of course this conjecture also include for bases other than 10, i.e. for every (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) family other than [URL="https://primes.utm.edu/glossary/xpage/GeneralizedRepunitPrime.html"]generalized repunit[/URL] family (b^n-1)/(b-1) (i.e. a=1 and c=-1), there are infinitely many primes not dividing any number of the form (a*b^n+c)/gcd(a+c,b-1), e.g. for the base 11 unsolved family 5{7} = (57*11^n-7)/10, the primes dividing no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (57*11^n-7)/10 are 3, 7, 11, 19, 37, 43, 61, 83, 89, 107, 131, 137, 157, 191, 193, 199, 211, 229, 241, 257, 269, 307, 311, 313, 317, 379, 389, 397, 421, 431, 439, 449, 457, 479, 503, 509, 521, 523, 541, 547, 571, 577, 607, 617, 631, 641, 653, 659, 661, 691, 727, 739, 743, 751, 757, 773, 787, 797, 811, 827, 829, 907, 911, 919, 967, ..., and for the base 17 unsolved family F1{9} = (4105*17^n-9)/16, the primes dividing no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (4105*17^n-9)/16 are 3, 5, 17, 29, 43, 59, 67, 71, 79, 101, 103, 137, 151, 157, 163, 179, 181, 191, 199, 223, 229, 239, 241, 257, 263, 281, 293, 307, 331, 337, 353, 359, 373, 383, 389, 409, 433, 443, 457, 461, 463, 491, 509, 541, 563, 587, 601, 619, 631, 647, 659, 661, 727, 733, 739, 757, 761, 769, 773, 797, 811, 821, 829, 859, 863, 877, 883, 919, 937, 947, 953, 967, 977, 991, ... (it is surprising that many primes do not divide (4105*17^n-9)/16 for any n, since (4105*17^n-9)/16 is a low-[URL="https://www.rieselprime.de/ziki/Nash_weight"]weight[/URL] form, i.e. (4105*17^n-9)/16 is divisible by a small prime for most n, note that (4105*17^n-9)/16 has no algebraic factorization for any n, since 4105 is not perfect power)

sweety439 2021-10-16 13:54

The smallest generalized near-repdigit primes (i.e. of the form x{y} or {x}y) base b is always minimal primes (start with b+1) in base b unless the repeating digit (i.e. y for x{y}, or x for {x}y) is 1, since the generalized repunit numbers base b may be prime unless b is 9, 25, 32, 49, 64, 81, 121, 125, 144, ... ([URL="https://oeis.org/A096059"]A096059[/URL]) bases without any generalized repunit primes, and for a repdigit to be prime, it must be a repunit (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7, thus (i.e. of the form x{y} or {x}y) base b is always minimal primes (start with b+1) in base b if the repeating digit (i.e. y for x{y}, or x for {x}y) is not 1, thus, the families A{1} in base 22 and 8{1} in base 33 and 4{1} in base 40 are not unsolved families in this problem (i.e. finding all minimal primes (start with b+1) in base b) although all they are near-repdigit families and all they have no known primes or PRPs and none of them can be ruled out as only contain composites (only count numbers > base), since their repeating digit are 1, and the prime F(1^957) in base 24 (its value is (346*24^957-1)/23) is not minimal prime (start with b+1) in base b=24, since its repeating digit is 1

sweety439 2021-10-17 17:00

The algebra form ((a*b^n+c)/d) of the unsolved families are:

[CODE]
base unsolved family algebra form
11 5(7^n) (57*11^n-7)/10
13 9(5^n) (113*13^n-5)/12
13 A(3^n)A (41*13^(n+1)+27)/4
16 (3^n)AF (16^(n+2)+619)/5
16 (4^n)DD (4*16^(n+2)+2291)/15
[/CODE]

See [URL="https://stdkmd.net/nrr/exprgen.htm"]https://stdkmd.net/nrr/exprgen.htm[/URL] for the algebra form calculator (only for base 10 families), also see page 16 of [URL="https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf"]https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf[/URL]

sweety439 2021-10-18 18:04

Families which can be ruled out as contain no primes (only count numbers > base) by reasons other than trivial 1-cover are:

[CODE]
Base 5:
{1}3 (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
{1}4 (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
3{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
4{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
Base 8:
1{0}1 (sum of cubes)
6{4}7 (covering set {3,5,13}) (not produce minimal primes (start with b+1) since 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 is prime)
Base 9:
{1} (difference of squares)
{1}5 (covering set {2,5})
2{7} (covering set {2,5})
3{1} (difference of squares)
{3}5 (covering set {2,5})
3{8} (difference of squares)
{3}8 (covering set {2,5})
5{1} (covering set {2,5})
5{7} (covering set {2,5})
6{1} (covering set {2,5})
{7}2 (covering set {2,5})
{7}5 (covering set {2,5})
{8}5 (difference of squares)
Base 11:
{1}3 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}4 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}9 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}A (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
2{5} (covering set {2,3})
3{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
3{5} (covering set {2,3})
3{7} (covering set {2,3})
4{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
4{7} (covering set {2,3})
{5}2 (covering set {2,3})
{5}3 (covering set {2,3})
{5}8 (covering set {2,3})
{5}9 (covering set {2,3})
{7}4 (covering set {2,3})
{7}9 (covering set {2,3})
{7}A (covering set {2,3})
8{5} (covering set {2,3})
9{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
9{5} (covering set {2,3})
9{7} (covering set {2,3})
A{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
A{7} (covering set {2,3})
Base 12:
{B}9B (combined with covering set {13} and difference of squares)
[/CODE]


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