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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] |
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] |
[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. |
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) |
[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 |
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) |
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Update pdf file for the proofs (not complete, continue updating ....)
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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. |
* 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. |
* 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. |
* 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. |
* 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. |
* 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) |
[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}) |
Have you thought about writing a book about all of this exciting stuff?
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14 posts on a single topic in 36 hours. Are you trying to get banned? It's working.
|
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[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 |
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). |
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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 |
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 |
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=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 |
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] |
There are totally 106 minimal primes (start with 2 digits) in base 12, there are 77 such primes in base 10
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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 |
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. |
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
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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. |
[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 |
[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) |
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 |
[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 |
[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. |
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[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. |
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See [URL="https://github.com/xayahrainie4793/non-single-digit-primes"]https://github.com/xayahrainie4793/non-single-digit-primes[/URL] for more data. |
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)] |
[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 |
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]) |
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) |
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) |
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] |
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] 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[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] 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[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] |
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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 |
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] |
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
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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.
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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 |
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. |
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] |
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] |
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 * |
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) |
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) |
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 |
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]) |
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] |
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) |
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]
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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] |
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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] |
[URL="https://sites.google.com/view/minimal--primes"]https://sites.google.com/view/minimal--primes[/URL]
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Update the data text file.
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[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] |
* 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. |
* 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. |
[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) |
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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) |
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). |
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Newest pdf files attached. Especially, base 16 all primes with <=9 digits are searched.
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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)
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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] |
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 |
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. |
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, 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F003DD0D, F00555A7, F005A557, F00C0D0D, F00CCCD9, F00D0D93, F030D0DD, F0555557, F070070A, F077077A, F0B00007, F0CC0D0D, F0CCB00D, F0D00DD9, F0DD0D0D, F0DDDDDF, F0FFFA6F, F300033D, F3000997, F33DDD4D, F40000CC, F4DDDDAA, F5A55575, F77007A7, F770707A, F770770A, F777007A, F77A7777, F77F0005, FA4AAAA4, FA4AAAAA, FA6000FF, FAAAAA44, FAAAAFF4, FB000003, FB000B33, FB330003, FBB00B0B, FBBB000B, FBBB0BBB, FBBB303F, FC003DDD, FCCCC9CD, FCCCCCD9, FCCCCDD4, FCDD0D0D, FCDD1003, FCFFFFFB, FD000DDD, FD0DD00D, FD0DD777, FD0DDDF3, FD330007, FD7777A7, FDDD000D, FDDDDA17, FDDDDD7A, FDDDDDA7, FF000C05, FFAAAF4A, FFAFFF4A, FFBBB303, FFF0A066, FFF3CC34, FFFAAA4A, FFFAAAA4, FFFB0333, FFFB3F03, FFFCF005, FFFCF555, FFFCFBBB, FFFCFF9B, FFFCFFB3, FFFF00C5, FFFFCBBB, FFFFFAA4, FFFFFCB3, FFFFFFC5, G0GGGGG1, G3033303, G6666FFF, G66FCCCC, G6FCCCCC, G6FFCFFF, G7000202, G7077772, G7077EEE, G77777E4, G77777FD, G7777DD3, G7777E74, G7EEE444, GAAAAFDF, GAAAGF66, GCFCFFFF, GDDDD4E7, GF077776, GF66CFFC, GFF66FFF, GGG0G333, GGG33333, GGG33366, GGG6CFFF, GGGFCCFF, GGGGG113, 100000FA5, 10000A01F, 10000DA01, 10000FA05, 100070009, 10009000D, 100109998, 10030000F, 100FA0005, 101999998, 107000009, 10AFFFFF5, 17707000F, 17F777757, 1F0A00005, 1F7777757, 1FFFFFF99, 2000B0B0B, 2005BBB0B, 200BB000B, 20B0000BB, 20B555555, 20BB0000B, 20BBB222E, 22222228B, 22222B222, 2AAAAAA6F, 2B5BBBBB2, 2BB0B00BB, 2BB0BB00B, 2BBB00B0B, 2EEEEEE6F, 300000404, 300000D74, 30004DDDD, 3000D4DDD, 300FFFFF5, 30D000001, 30D000DDD, 30D400DDD, 30DDDD747, 30F0BBBBB, 31000000F, 333333395, 3333333FD, 3333335DD, 33333377D, 333339995, 33333C305, 337444444, 340DDDDDD, 35000000D, 355555553, 355555595, 399955555, 3BBBB333D, 3BBBBBB1B, 3CCCCFCCD, 3CCFBBBBB, 3DD4000DD, 3E6000F0F, 3FCCCCCCD, 40000100G, 400001A0G, 4000AAAAD, 40010000A, 409DDDDDD, 40DDDDDD9, 40DDDDDEE, 41G444444, 444666669, 444699999, 44EEE9909, 4D0000D0D, 4D0GEEEEE, 4DA777777, 4DAD77777, 4DD0DDDDD, 4DDDDD11A, 4DDDEEAAA, 50000003B, 50000010D, 500000701, 500001FFF, 50000570A, 500005FB7, 50000D009, 50005557A, 500150A0A, 500F0055B, 5011FFF0F, 50570000A, 507000005, 50700010A, 509000005, 50B000009, 535BBBBBB, 555550305, 555555809, 55A5AA55E, 55AAA5AAE, 585555505, 588555595, 588858555, 589000007, 58A555555, 58A888855, 58AAA5555, 58AAAAA55, 58E885555, 5A5AAA5AE, 5A7000005, 5AA5A5552, 5BBB3DBBB, 5BBBBBBDF, 5D0000009, 5DDD00007, 5DDD00009, 5DDDDDE07, 5FFFF5FFB, 5FFFFF5BF, 60000999G, 60066999G, 606666E96, 63633333D, 666666698, 666666E96, 666669GFF, 6666CGGG3, 66G033333, 66GGG3033, 6999999GF, 69G333333, 6AAADD00F, 6ADD0000F, 6CC00CCE9, 70000021B, 700000B92, 700007005, 70000770A, 70000B911, 700090177, 700097002, 7000A7777, 7000D03EE, 70077000A, 700770DDA, 700900001, 700A7000F, 700F0A001, 70700007A, 707077E7A, 707770005, 7077770A2, 7077777E5, 7077A7777, 709000001, 70D00F0DD, 70D0B00EE, 70DD0E0EE, 70DF0D00D, 70F077777, 7200000B1, 740000D0D, 742000002, 7444G4444, 744G44444, 747099999, 747999909, 75DF0000D, 760999999, 77000000D, 7700000EF, 77000070A, 77000707F, 77000EEE9, 7700700DD, 7700D00FD, 770700D0D, 77070700F, 770707DDF, 770D000DD, 77400000D, 777000DDA, 77707777A, 777400D0D, 777770A77, 777777496, 7777775D9, 77777772E, 777777797, 7777777F7, 777777D9F, 777779007, 77777D409, 77777E479, 7777EE409, 7777F7005, 7777G7703, 777DF0D0D, 777F0000D, 777F77AD4, 77D0000DD, 77DD000EF, 77DD44409, 77E77777A, 77E777907, 77E7E7779, 77EEEEEG7, 791199999, 799999009, 7A4G44444, 7D00DD03E, 7D0B00E0E, 7D0D0000E, 7D0DEEEEE, 7D0EBE0EE, 7DD00D03E, 7DDEEE0EE, 7E7EE00EF, 7EEEE444G, 7EEEEEE4A, 7EEG70777, 7EG070777, 7F00000A5, 7F7777757, 7G7777747, 7G77777F6, 80000005A, 800000074, 800000085, 800008E85, 800050075, 805555005, 809555555, 80EEEEEEB, 855055555, 855555905, 855590555, 855900055, 858555595, 85A500005, 85AAA5555, 88AAA5556, 8B2E0000E, 8B8555555, 8CCBEBBBB, 8CCCCC096, 8CCEBBBBB, 900000736, 900070333, 9000D0DDD, 905525555, 90777DD0D, 909900905, 909C55555, 918000007, 919999995, 92222222G, 95555555F, 962222222, 96C666662, 97770000F, 977733003, 977777333, 977777775, 97777777F, 986606666, 986660006, 986660666, 986666006, 990000005, 990000959, 991777777, 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) |
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}: 95906589091783507930843563755990266422092688356040675515311603708093961802306894007437655217514330623235207811600088376616459742526084918320106897318128806495781475426768767165514390697631437051279309375222665888996529269923522513150777949332557515521324424371314404893207152062522816208660683728587344752060207752176117852541701300415836483279179797153360991805611346793920572915966693512831758798610318187165078100052657265407064700184425873728839407610370409154466626422741733541020144469899847293106180159054565336330268644842349049592211776064686938080891658404158572628900407902382793468029662463581407704575520153279074959618731284192450532813079007371781177428028956685990102815330642281362082111577053562596575982200756771882498844269795552481099163314315787859968104394148733642046151581817378168289655730735285607627344702459668007180874848122337772142808275765179310807830012018795922800290688494743524724295147812117289642950083881206753277901109889820750317055789894070812884761502938943605687373479513650470502061201712723954528850575691371252234690260912366780751178462203664771090103741876657991281157114168815982359929638882464261198071565661897533385653118572545290080553260068584635774174378873241967998873902779616049492290819388713047447836299281834501905197895513178008430148838039753653476431746091649918729698629758813644179453577167011361866263064663482484971061912954834835200477371046714024946618399066857015486252167195960074052560082946015310807355599115365154898206253970363795413248288426431043732650612453678289003108049110516734699599633783987864173701209615158542221041250911073433615334565034714121771220193160128195806230453409786749742845939608758929141872445533757921260002927508261714697148904081140693515371012423699033127330493467490048678198295121841737113578761601362608451647399392411505661604089075312833905939420635095935065291556854483070663161964127222184661472915798063325075795310241827209806620530513237119860789944360183319440790216620845715705884866599976338449612477998170052792542193549653671825352068259451901384834348260380145635795902195520839603052665037992217129182586156768151478332476293754270819051535328015556313203975988535607477203174867043926355262515269619989893528810931352894985781863647407409963294078899410144835689753338000750769507629898619664022834082535232488316695619138887782288109959951826676589454631927024639295210111806877585535436438777573680792913296936989667348185106551317537036603215416045634454099384405880405696970720084684693247251343705527039783036991738239849876055320891925638823902862780492307005254189057301401817013458964613400020256906005181311284022352648807314852064365215297536521853957702230484874643069416504295038457867649364475841900123618490276713195796005557064624890834770622094022000187190171714522228753474852925074066174542928946709438848843012663738859945859221737289291171096569517311534969894552062670173368644363541663192337912778284843268514815921286383827432399361150667678322987438015028340370193694146594687267967014425498222196335366572333217884787193471118715216854187874189418504437464870873131183472181208042966524528356295092674006998162392804478553820447794088341938629744582995866448719357944154114949377566255938868506935999098594262466142856156343589113789200203891385805977216641210094514355114576835097945155598461070241035469794842774507531812654666380339725829035457927631544810648512697742475840941044313592470433686717439064577986319891517211466466148642266977218462335259214786775338256474834203607752993286655331560107205540478606754476315314952772003753859459618476201332195124702019538036820429400127713096356869469174745389203770244409167126770249782096482566537299311052679852899076507988153936077495361427149560460699617095838241560999866892201345400745108331228104164320135716700859602712441769263379666713760107563730203796881758124746645149986594048960910932208225161109728474522151908665166819659957697315780018637157866623837740131680236973888742969842123536647664781356394353599098937132120578421177881953953489697403221141910722457903659248285030041952003816703834491628113647737078152275019652593835963898544432294235176334662225559536463552759541484261645409937761720728736185109624513914201497900199535439579294128406730426260799860203048262445526683425847539709798431034743757191922233526369259328361286021444496038217776974673610871233162722242391857718914109493890692932421845551321497560276396591188063845546634526087874534643700005680620069287168146621038580084350836816579510989510482628397457066953288858121565062507610370561970649072966964965050553424650069839765408206450820404370870725339068554206739587431061242394089506855550109461263470386002891976348462610586433717527777170636075792769147581795163223229658353536650742052756349958237927267250696531810315269126535665858820819115046643587993989863564977869489984571848082751410762948010913445945171202057182212644951369452552196946306607657770384335079326676617600668092435117966493471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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] |
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: 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A{0}F8: 49393 17: AF{0}C: 262673557 17: A{0}FC: 165179769267805060028338008295315840289767273632277 17: A{0}FE: 14198839 17: A{0}FG: 671328806001012829487353559941943176207647465878611669861215642487108848281 17: A{0}G5: 3167 17: A{0}GF: 49417 17: B1{0}1: 22294520781437 17: B{0}11: 1852156092153410237 17: B1{0}3: 15701951 17: B{0}13: 918751 17: B1{0}5: 4537862977 17: B1{0}7: 3203 17: B{0}17: 3203 17: B1{0}9: 923653 17: B{0}19: 3733935384460730951078545288357 17: B1{0}B: 4537862983 17: B{0}1B: 219643257909454761828149722871 17: B1{0}D: 3209 17: B{0}1D: 3209 17: B1{0}F: 54347 17: B{0}1F: 918763 17: B3{0}1: 4586138111 17: B{0}31: 15618479 17: B3{0}3: 4586138113 17: B{0}33: 108950358361965361 17: B{0}35: 918787 17: B3{0}7: 54917 17: B{0}37: 54101 17: B3{0}9: 54919 17: B{0}39: 265513319 17: B3{0}B: 13127467443471659247357171241 17: B{0}3B: 918793 17: B3{0}D: 0 17: B{0}3D: 4513725467 17: B5{0}1: 111863469548114113 17: B{0}51: 6408844609527457 17: B5{0}5: 943301 17: B{0}55: 54133 17: B5{0}7: 3271 17: B{0}57: 3271 17: B{0}59: 520095817114761636162014143632871122816179525449726864640127285167417 17: B5{0}B: 943307 17: B{0}5B: 54139 17: B5{0}D: 55501 17: B{0}5D: 918829 17: B{0}5F: 15618527 17: B7{0}1: 3299 17: B{0}71: 3299 17: B7{0}3: 3301 17: B{0}73: 3301 17: B7{0}5: 4682688391 17: B{0}75: 54167 17: B7{0}7: 120368467402913744569952911790435540211109602316792037467773439161424642184660205394353665329205357023441188929696281728207111532278907384327215756322802584162406684848144505707655328848002179437362480355620106618551759942953072098298230080760730334893892256109012808126702757356801365677371486449935135081516636221666088529070482165198728725452761709696965018389084137580053513363934803454848623559742453151893033043489938889027709341248431944645241366452532317174972241937258171643460574677275700486569 17: B{0}77: 918857 17: B7{0}9: 3307 17: B{0}79: 3307 17: B7{0}B: 4682688397 17: B{0}7B: 15618557 17: B7{0}D: 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17: BF{0}1: 58379 17: B{0}F1: 4513725659 17: BF{0}3: 992429 17: B{0}F3: 918989 17: BF{0}5: 252668780465541316571672556099331582114159072512820097741526544331987941050321324854188949616565082693594564109779103002724156221108775977419041756841765029191992489406827190551736970531991443526838126560523171683941893586079748831261355888354129351713544303608009104113641686764242295791916122562065053595252331127040041747158845205462470772521844595935311089249604050120511233621228991727537404168839942406816757142997798537795276019942609964930304230887731954804066232633673386929979790343169113007757877964091759 17: B{0}F5: 76733332111 17: BF{0}7: 9829560758884707433169 17: B{0}F7: 63476901535832426168335269901889 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] |
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] |
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] |
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]) |
2 Attachment(s)
Newest data (minimal primes (start with b+1) up to certain limit) for bases b=17 and b=18
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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] |
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. |
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]. |
1 Attachment(s)
upload the text file for the proof for bases 2, 3, 4, 5, 6, 8, 10, 12
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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.
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[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. |
[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. |
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.
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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, ... |
[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. |
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. |
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] |
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) |
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
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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] |
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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