20201030, 23:05  #45 
Nov 2016
2,819 Posts 
Unsolved families which are CRUS Sierpinski/Riesel problems but with k's > CK:
Base 32: G{0}1: 16*32^n+1 UG{0}1: 976*32^n+1 Base 33: FFF{W}: 16846*33^n1 Base 41: FZ{0}1: 650*41^n+1 R0R8{0}1: 1861982*41^n+1 S{0}1: 28*41^n+1 XL4{0}1: 56338*41^n+1 Z098{0}1: 2412612*41^n+1 Z0R{0}1: 58862*41^n+1 EF{e}: 590*41^n1 PI{e}: 1044*41^n1 UFM{e}: 51068*41^n1 UX{e}: 1264*41^n1 XOC{e}: 56470*41^n1 XQO{e}: 56564*41^n1 XQXXXX{e}: 3899055672*41^n1 XQ{e}: 1380*41^n1 Base 43: Y6{0}1: 1468*43^n+1 6XF{0}1: 12528*43^n+1 8Q6{0}1: 15916*43^n+1 XZZ{g}: 62558*43^n1 YFa{g}: 63548*43^n1 dcU{g}: 73776*43^n1 4ZZZ{g}: 384284*43^n1 8OR{g}: 15852*43^n1 9QQ{g}: 17786*43^n1 FFFFFFFQ{g}: 4174357242012*43^n1 FFFQ{g}: 1221012*43^n1 
20201030, 23:48  #46 
Nov 2016
2,819 Posts 
All smallest generalized repunit prime base b are minimal prime base b, since they are of the form {1} in base b, for the smallest generalized repunit (probable) prime base b for b<=1024, see https://raw.githubusercontent.com/xa...iesel%20k1.txt
All smallest generalized Fermat prime base b (for even b) and all smallest generalized half Fermat prime base b (for odd b) are minimal prime base b, unless (b1)/2 is prime for odd b, since they are of the form 1{0}1 in base b (for even b) or {(b1)/2}(b+1)/2 in base b (for odd b), for the smallest generalized (half) Fermat (probable) prime base b for b<=1024, see https://raw.githubusercontent.com/xa...Sierp%20k1.txt 
20201031, 00:02  #47 
Nov 2016
2,819 Posts 
There are no known generalized repunit (probable) primes in these bases <= 1024: (search limit: 100000)
{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} There are no known generalized (half) Fermat (probable) primes in these bases <= 1024: (search limit: 2^22 for GFN for even bases, 2^18 for half GFN for odd bases) {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 179, 182, 183, 186, 189, 197, 200, 202, 207, 211, 212, 214, 215, 218, 223, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 281, 283, 285, 286, 287, 291, 293, 294, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 344, 347, 351, 354, 355, 356, 359, 362, 367, 368, 369, 377, 380, 383, 387, 389, 390, 394, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 458, 467, 468, 469, 473, 475, 480, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 514, 515, 518, 524, 528, 530, 533, 534, 538, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 622, 626, 627, 629, 632, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 675, 678, 679, 683, 684, 687, 691, 692, 694, 698, 706, 707, 709, 712, 720, 722, 724, 731, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 759, 762, 766, 767, 770, 771, 773, 775, 783, 785, 787, 792, 794, 797, 802, 806, 807, 809, 812, 813, 814, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 867, 868, 870, 872, 873, 878, 887, 888, 889, 893, 896, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1002, 1003, 1006, 1009, 1014, 1016} Last fiddled with by sweety439 on 20201031 at 00:03 
20201031, 00:05  #48  
Nov 2016
2,819 Posts 
Quote:
{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} The half GFN for these bases are also minimal primes: {31, 55, 67, 77, 89, 91, 97, 99, 109, 127, 137, 149, 151, 155, 161, 183, 189, 197, 211, 223, 233, 235, 241, 247, 249, 257, 265, 269, 281, 283, 285, 287, 291, 293, 307, 311, 319, 351, 355, 367, 369, 377, 389, 401, 407, 411, 413, 417, 437, 439, 443, 469, 473, 475, 489, 493, 495, 497, 509, 511, 533, 547, 549, 591, 593, 597, 601, 603, 611, 619, 629, 637, 645, 647, 651, 653, 655, 659, 667, 671, 679, 683, 687, 691, 709, 731, 737, 741, 743, 749, 753, 755, 771, 773, 775, 783, 785, 787, 797, 807, 809, 813, 823, 825, 849, 851, 853, 873, 889, 893, 903, 907, 911, 937, 939, 941, 943, 945, 947, 953, 961, 967, 977, 987, 993, 1003, 1009} However, the half GFN for these bases are not minimal primes, since (b1)/2 is prime: {63, 83, 107, 123, 135, 143, 147, 179, 207, 215, 227, 255, 263, 303, 327, 347, 359, 383, 387, 423, 447, 467, 483, 515, 555, 563, 615, 627, 635, 663, 675, 707, 735, 759, 767, 867, 887, 899, 915, 923, 927, 975, 983, 999} 

20201031, 00:09  #49 
Nov 2016
2,819 Posts 
The remain k < b (also including k > CK, if k < b, e.g. 28*41^n+1 and 27*34^n1) in CRUS corresponding to minimal primes to base b if and only if:
* In Sierpinski case, k is not prime (since it is k{0}1 in base b) * In Riesel case, neither k1 nor b1 is prime (since it is (k1){(b1)} in base b) e.g. the smallest prime of the form 4*53^n+1 (already searched to 1.65M) will be minimal prime base 53, if it exists (CRUS conjectured that they all exist) Last fiddled with by sweety439 on 20201031 at 00:11 
20201109, 18:15  #50  
Nov 2016
2,819 Posts 
Quote:
* Cubes: ** Base 8: GFN in base 2 are either 2{0}1 or 4{0}1 in base 8, however, 2 and 401 are primes, thus, base 8 does not have GFN or half GFN remain. ** Base 27: half GFN in base 3 are either 1{D}E or 4{D}E in base 27, however, D is prime, thus, base 27 does not have GFN or half GFN remain. ** Base 64: GFN in base 2 are either 4{0}1 or G{0}1 in base 64, however, 41 and G01 are primes, thus, base 64 does not have GFN or half GFN remain. ** Base 125: half GFN in base 5 are either 2:{62}:63 or 12:{62}:63 in base 125, however, 2 is prime, but the family 12:{62}:63 does not have any known (probable) prime (the only known half GFN (probable) primes in base 5 are 3, 13, 2:63), thus, base 125 has half GFN remain. ** Base 216: GFN in base 6 are either 6:{0}:1 or 36:{0}:1 in base 216, however, 6:1 is prime, but the family 36:{0}:1 does not have any known prime (the only known GFN primes in base 6 are 7, 37, 6:1), thus, base 216 has GFN remain. ** Base 343: half GFN in base 7 are either 3:{171}:172 or 24:{171}:172 in base 343, however, 3 is prime, but the family 24:{171}:172 does not have any known (probable) prime (the only known half GFN (probable) prime in base 7 is 3:172), thus, base 343 has half GFN remain. ** Base 512: GFN in base 2 are 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 128:{0}:1, or 256:{0}:1 in base 512, however, 2 and 128:1 are primes, but the families 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 128:1), thus, base 512 has GFN remain. ** Base 729: half GFN in base 3 are either 4:{364}:365 or 40:{364}:365 in base 729, however, 40:364:365 and 4:364:364:364:364:365 are primes, thus, base 729 does not have GFN or half GFN remain. ** Base 1000: GFN in base 10 are either 10:{0}:1 or 100:{0}:1 in base 1000, and both families do not have any known prime (the only known GFN primes in base 10 are 11 and 101), thus, base 1000 has GFN remain. * 5th powers: ** Base 32: GFN in base 2 are 2{0}1, 4{0}1, 8{0}1, or G{0}1 in base 32, however, 2 and 81 are primes, but the families 4{0}1 and G{0}1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, H, 81, 2001), thus, base 32 has GFN remain. ** Base 243: half GFN in base 3 are 1:{121}:122, 4:{121}:122, 13:{121}:122, or 40:{121}:122 in base 243, however, 1:121:121:122, 4:121, 13, 40:121:121:121:121:121:121:121:121:121:121:121:122 are primes, thus, base 243 does not have GFN or half GFN remain. ** Base 1024: GFN in base 2 are 4:{0}:1, 16:{0}:1, 64:{0}:1, or 256:{0}:1 in base 1024, however, 64:1 is prime, but the families 4:{0}:1, 16:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 64:1), thus, base 1024 has GFN remain. * 7th powers: ** Base 128: GFN in base 2 are 2:{0}:1, 4:{0}:1, or 16:{0}:1 in base 128, however, 2 and 4:0:1 are primes, but the family 16:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 2:1, 4:0:1), thus, base 128 has GFN remain. 

20201109, 18:21  #51 
Nov 2016
2,819 Posts 
There are about exp(gamma*k) minimal primes in base n, where k = number of 2digit numbers xy in base n such that none of x, y, xy are primes, x != 0, gcd(y,n) = 1
where exp(x) = e^x (e is the base of the natural logarithm (2.718281828...), gamma is Eulerâ€“Mascheroni constant (0.5772156649...)) Also, there are about exp(gamma*k) minimal strings of primes with >=2 digits in base n (see thread https://mersenneforum.org/showthread.php?t=24972), where k = number of 2digit numbers xy in base n such that xy is not prime, x != 0, gcd(y,n) = 1 Last fiddled with by sweety439 on 20201110 at 16:13 
20201110, 16:21  #52  
Nov 2016
2,819 Posts 
Quote:
The k for the original case (i.e. including the singledigit primes) Code:
base,k 2,0 3,1 4,0 5,4 6,1 7,5 8,3 9,8 10,5 11,27 12,2 13,38 14,10 15,23 16,17 17,84 18,4 19,108 20,17 21,59 22,30 23,164 24,9 25,151 26,57 27,136 28,55 29,307 30,8 31,350 32,87 33,190 34,111 35,282 36,42 37,539 38,144 39,289 40,107 41,678 42,31 43,736 44,169 45,295 46,227 47,892 48,59 49,804 50,160 51,543 52,286 53,1194 54,85 55,842 56,284 57,731 58,416 59,1545 60,47 61,1627 62,464 63,738 64,508 65,1248 66,144 67,2031 68,537 69,1101 70,265 71,2296 72,190 73,2404 74,676 75,936 76,696 77,1943 78,203 79,2867 80,503 81,1623 82,912 83,3179 84,150 85,2275 86,999 87,1865 88,911 89,3750 90,110 91,2865 92,1121 93,2182 94,1285 95,3009 96,456 97,4603 98,1012 99,2249 100,901 101,4994 102,420 103,5158 104,1347 105,1500 106,1635 107,5562 108,539 109,5725 110,812 111,3123 112,1300 113,6178 114,502 115,4391 116,1852 117,3231 118,2048 119,5209 120,273 121,6478 122,2286 123,4081 124,2313 125,5810 126,536 127,8241 128,2568 Code:
base,k 2,0 3,2 4,2 5,10 6,2 7,25 8,14 9,30 10,15 11,75 12,15 13,111 14,40 15,70 16,72 17,202 18,43 19,260 20,82 21,163 22,126 23,394 24,88 25,375 26,187 27,348 28,196 29,648 30,88 31,749 32,335 33,470 34,348 35,627 36,221 37,1089 38,450 39,684 40,385 41,1350 42,231 43,1495 44,579 45,764 46,685 47,1802 48,425 49,1674 50,628 51,1237 52,846 53,2311 54,549 55,1742 56,891 57,1575 58,1138 59,2894 60,458 61,3099 62,1316 63,1701 64,1470 65,2512 66,724 67,3766 68,1539 69,2370 70,1021 71,4245 72,1034 73,4500 74,1927 75,2242 76,1964 77,3802 78,1076 79,5295 80,1716 81,3495 82,2395 83,5861 84,1109 85,4476 86,2654 87,3879 88,2521 89,6768 90,1142 91,5466 92,2970 93,4467 94,3202 95,5671 96,1922 97,8078 98,2914 99,4697 100,2756 101,8774 102,1984 103,9137 104,3656 105,3683 106,4130 107,9883 108,2480 109,10270 110,2942 111,6478 112,3859 113,11051 114,2551 115,8490 116,4876 117,6765 118,5170 119,9691 120,2152 121,11515 122,5547 123,8024 124,5614 125,10609 126,2682 127,14030 128,6259 

20201110, 16:28  #53  
Nov 2016
5403_{8} Posts 
Quote:
Code:
base number of unsolved families when searched to 10000 digits 17 2 34 33 19 5 38 77 21 3 42 0 (the largest prime has only 487 digits) 

20201110, 18:13  #54  
Nov 2016
2,819 Posts 
Quote:


20201125, 08:11  #55 
Nov 2016
5403_{8} Posts 

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