A Sierpinski/Riesel base b has 2-cover if and only if b+1 is divisible by at least two distinct odd primes, such bases b are
- 12, 18, 25, 28, 2X, 32, 35, 38, 42, 46, 48, 4E, 52, 54, 55, 58, 59, 62, 64, 65, 6E, 70, 72, 75, 76, 78, 7X, 82, 85, 88, 91, 92, 95, 96, 98, 9X, 9E, X2, X5, X8, X9, XE, E0, E2, E5, E7, E8, EX, 100, 102, 105, 108, 109, 10X, 10E, 112, 114, 118, 11E, 121, 122, 125, 126, 128, 12E, 131, 132, 134, 135, 136, 138, 139, 142, 145, 148, 14X, 14E, 150, 152, 154, 155, 158, 15X, 160, 162, 163, 164, 165, 168, 16E, 171, 172, 175, 176, 178, 179, 17E, 184, 185, 186, 188, 18E, 190, 192, 195, 196, 197, 198, 19E, 1X0, 1X1, 1X2, 1X5, 1X8, 1XX, 1XE, 1E2, 1E3, 1E5, 1E8, 1E9, 1EX, 201, 202, 205, 206, 208, 20X, 20E, 210, 212, 214, 215, 217, 218, 219, 21E, 222, 225, 226, 228, 229, 22X, 230, 232, 234, 235, 238, 23X, 23E, 242, 243, 244, 245, 248, 24E, 251, 252, 255, 256, 258, 25E, 262, 263, 264, 265, 268, 269, 26X, 26E, 271, 272, 274, 275, 277, 278, 280, 282, 285, 286, 288, 28X, 28E, 292, 295, 296, 298, 299, 29X, 29E, 2X1, 2X2, 2X4, 2X5, 2X6, 2X8, 2X9, 2XE, 2E2, 2E4, 2E5, 2E6, 2E8, 2E9, 301, 302, 304, 305, 307, 308, 309, 30E, 310, 312, 315, 316, 318, 31X, 31E, 322, 323, 325, 328, 32E, 330, 331, 332, 334, 335, 336, 337, 338, 33E, 340, 342, 344, 348, 349, 34E, 350, 351, 352, 354, 355, 358, 35E, 360, 361, 362, 365, 366, 368, 36X, 36E, 370, 371, 372, 373, 375, 378, 37X, 37E, 381, 382, 383, 384, 385, 386, 388, 38X, 38E, 392, 394, 395, 398, 399, 39X, 39E, 3X0, 3X2, 3X5, 3X6, 3X7, 3X8, 3XE, 3E0, 3E2, 3E5, 3E7, 3E8, 3E9, 3EX, 402, 403, 404, 405, 406, 408, 40E, 410, 411, 412, 415, 416, 418, 419, 41E, 421, 422, 424, 425, 428, 429, 42X, 42E, 432, 433, 435, 437, 438, 43X, 43E, 442, 444, 445, 448, 44X, 44E, 450, 451, 452, 455, 457, 458, 459, 460, 461, 462, 465, 466, 468, 469, 46E, 472, 474, 475, 476, 478, 479, 47X, 47E, 482, 485, 486, 487, 488, 489, 48E, 490, 492, 494, 495, 498, 49X, 49E, 4X0, 4X2, 4X3, 4X5, 4X6, 4X8, 4XX, 4XE, 4E1, 4E2, 4E4, 4E5, 4E6, 4E8, 4EE, 500, 502, 504, 505, 507, 509, 50X, 50E, 512, 514, 515, 517, 518, 519, 51E, 520, 522, 523, 524, 525, 528, 529, 52X, 52E, 532, 533, 535, 536, 538, 53X, 541, 542, 545, 546, 548, 54X, 54E, 550, 551, 552, 554, 555, 558, 559, 55X, 55E, 560, 562, 565, 566, 568, 56X, 56E, 570, 571, 572, 575, 577, 578, 579, 57X, 57E, 580, 582, 583, 585, 588, 589, 58E, 591, 592, 594, 595, 596, 597, 598, 59E, 5X2, 5X4, 5X5, 5X6, 5X8, 5X9, 5XX, 5XE, 5E1, 5E2, 5E5, 5E7, 5E8, 600, 602, 603, 604, 605, 606, 608, 609, 60X, 60E, 612, 613, 615, 617, 618, 61E, 620, 621, 622, 624, 625, 626, 628, 62X, 62E, 630, 631, 632, 634, 635, 638, 639, 63E, 640, 642, 644, 645, 647, 648, 64X, 64E, 650, 652, 655, 656, 658, 65X, 65E, 661, 662, 663, 665, 666, 668, 669, 66E, 670, 671, 672, 673, 675, 676, 678, 67X, 67E, 681, 682, 684, 685, 688, 689, 690, 692, 695, 696, 697, 698, 69E, 6X0, 6X1, 6X2, 6X3, 6X4, 6X5, 6X8, 6X9, 6XX, 6XE, 6E2, 6E4, 6E5, 6E6, 6E8, 6EX, 6EE, 701, 702, 703, 705, 706, 708, 70E, 711, 712, 714, 715, 716, 718, 719, 71E, 721, 722, 723, 724, 725, 727, 728, 72X, 72E, 730, 732, 735, 738, 739, 73X, 73E, 740, 742, 743, 745, 747, 748, 749, 74X, 74E, 751, 752, 754, 755, 756, 758, 759, 75X, 75E, 760, 762, 764, 765, 768, 769, 76E, 772, 775, 776, 777, 778, 779, 77E, 780, 781, 782, 785, 786, 788, 78X, 78E, 791, 792, 793, 794, 795, 798, 79X, 79E, 7X1, 7X2, 7X4, 7X5, 7X6, 7X8, 7XX, 7XE, 7E0, 7E2, 7E3, 7E4, 7E5, 7E6, 7E7, 7E8, 7E9, 802, 804, 805, 806, 807, 808, 809, 80E, 810, 811, 812, 814, 815, 818, 81X, 81E, 820, 821, 822, 823, 825, 826, 828, 82E, 830, 831, 832, 835, 836, 837, 838, 83X, 83E, 842, 843, 844, 845, 846, 848, 849, 84X, 84E, 852, 855, 856, 857, 858, 859, 85E, 860, 862, 865, 867, 868, 86X, 86E, 872, 873, 874, 875, 876, 878, 879, 87X, 87E, 882, 884, 885, 886, 888, 889, 88E, 890, 892, 894, 895, 896, 898, 899, 89X, 89E, 8X0, 8X1, 8X2, 8X3, 8X5, 8X8, 8XE, 8E0, 8E2, 8E3, 8E5, 8E7, 8E8, 8EX, 901, 902, 903, 905, 908, 90E, 910, 911, 912, 914, 915, 916, 917, 918, 91E, 922, 924, 925, 928, 929, 92E, 930, 931, 932, 934, 935, 936, 937, 938, 939, 93X, 93E, 940, 942, 944, 945, 946, 948, 94X, 94E, 950, 951, 952, 953, 955, 956, 957, 958, 95E, 961, 962, 965, 966, 968, 969, 96X, 96E, 972, 974, 975, 976, 978, 979, 97X, 97E, 980, 981, 982, 984, 985, 987, 988, 98X, 98E, 990, 991, 992, 995, 996, 998, 999, 99X, 99E, 9X0, 9X2, 9X3, 9X4, 9X5, 9X8, 9X9, 9XE, 9E1, 9E2, 9E5, 9E6, 9E8, 9EE, X00, X01, X02, X04, X05, X08, X09, X0E, X12, X13, X14, X17, X18, X19, X1X, X1E, X20, X22, X24, X25, X28, X29, X2X, X2E, X30, X32, X33, X35, X37, X38, X3E, X41, X42, X45, X46, X47, X48, X49, X4E, X50, X52, X54, X55, X56, X57, X58, X59, X5E, X60, X62, X64, X65, X66, X67, X68, X6E, X70, X71, X72, X74, X75, X78, X79, X7X, X80, X82, X83, X84, X85, X88, X8X, X8E, X91, X92, X95, X96, X98, X99, X9E, XX0, XX1, XX2, XX3, XX4, XX5, XX8, XX9, XXE, XE0, XE2, XE4, XE5, XE7, XE8, XE9, XEE, E00, E01, E02, E04, E05, E06, E08, E0X, E0E, E11, E12, E15, E16, E18, E19, E1E, E21, E22, E23, E25, E26, E28, E2E, E32, E33, E34, E35, E37, E38, E39, E3X, E3E, E40, E41, E42, E45, E46, E47, E48, E4X, E4E, E50, E52, E54, E55, E56, E57, E58, E5X, E5E, E62, E63, E64, E65, E68, E69, E6E, E71, E72, E73, E74, E75, E76, E78, E7X, E7E, E82, E84, E85, E86, E88, E89, E8X, E8E, E91, E92, E95, E97, E98, E99, E9X, E9E, EX0, EX2, EX3, EX5, EX6, EX8, EXX, EXE, EE0, EE2, EE3, EE5, EE8, EEX, 1000, ...
If b+1 is divisible by two distinct odd primes p and q, then all numbers = +1 mod p and −1 mod q, and all numbers = −1 mod p and +1 mod q, are both Sierpinski numbers and Riesel numbers to base b
For the status of such bases b≤100:
b | the odd primes dividing b+1 | conjectured smallest Sierpinski/Riesel number calculated by the condition | the true Sierpinski CK and the reason (blank if the true Sierpinski CK is the same as the CK calculated by the condition) | the true Riesel CK and the reason (blank if the true Riesel CK is the same as the CK calculated by the condition) | status of the Sierpinski conjecture base b and the k needing exponent > 1000 | status of the Riesel conjecture base b and the k needing exponent > 1000 |
12 | 3, 5 | 4 | proven | proven | ||
18 | 3, 7 | 8 | proven | proven | ||
25 | 3, 5 | 4 | proven | proven | ||
28 | 3, E | X | proven | proven | ||
2X | 5, 7 | 6 | proven | proven | ||
32 | 3, 11 | 12 | 11, since 11×32^{n}−1 has covering set {3, 5, 15} | proven (k = 2 has exponent n = 16E5) | proven | |
35 | 3, 7 | 8 | proven | proven | ||
38 | 3, 5 | 4 | proven | proven | ||
42 | 3, 15 | 14 | proven | proven | ||
46 | 5, E | 19 | proven | proven | ||
48 | 3, 17 | 18 | proven | proven | ||
4E | 3, 5 | 4 | proven | proven | ||
52 | 3, 7 | 8 | proven | proven | ||
54 | 5, 11 | 12 | 43, since gcd(12+1,54−1) is not 1, and 43 is the next k calculated by the condition | proven | proven | |
55 | 3, E | X | proven | proven | ||
58 | 3, 1E | 1X | k = 15 has no known prime (k = 10 has exponent 2781E5) | proven (k = 5 has exponent 7X32, k = 7 has exponent 12843) | ||
59 | 5, 7 | 6 | proven | proven | ||
62 | 3, 5 | 4 | proven | proven | ||
64 | 7, E | 2X | 37, since gcd(2X+1,64−1) is not 1, and 37 is the next k calculated by the condition | X0, since gcd(2X−1,64−1) is not 1, and 37 is the next k calculated by the condition, and gcd(37−1,64−1) is not 1, and 93 is the next k calculated by the condition, and gcd(93−1,64−1) is not 1, and X0 is the next k calculated by the condition | proven | proven |
65 | 3, 11 | 12 | proven (k = 4 has exponent 3642) | proven | ||
6E | 3, 7 | 8 | proven (k = 4 has exponent 3492) | proven | ||
70 | 5, 15 | 14 | proven | proven | ||
72 | 3, 25 | 24 | k = 8 has no known prime | proven | ||
75 | 3, 5 | 4 | proven | proven | ||
76 | 7, 11 | 23 | proven | proven | ||
78 | 3, 27 | 28 | proven | proven | ||
7X | 5, 17 | 33 | proven | k = 25 has no known prime | ||
82 | 3, E | X | proven | proven | ||
85 | 3, 15 | 14 | 9X, since gcd(14−1,85−1) is not 1, and 2E is the next k calculated by the condition, and gcd(2E−1,85−1) is not 1, and 57 is the next k calculated by the condition, and gcd(57−1,85−1) is not 1, and 72 is the next k calculated by the condition, and gcd(72−1,85−1) is not 1, and 9X is the next k calculated by the condition | proven (k = 2 has exponent 9332E) | proven (k = 58 has exponent 316X, k = 90 has exponent 14480) | |
88 | 3, 5, 7 | 4 | proven | proven | ||
91 | 5, E | 19 | 2X, since gcd(19+1,91−1) is not 1, and 2X is the next k calculated by the condition | 100, since gcd(19−1,91−1) is not 1, and 2X is the next k calculated by the condition, and gcd(2X−1,91−1) is not 1, and 64 is the next k calculated by the condition, and gcd(64−1,91−1) is not 1, and 75 is the next k calculated by the condition, and gcd(75−1,91−1) is not 1, and XE is the next k calculated by the condition, and gcd(XE−1,91−1) is not 1, and 100 is the next k calculated by the condition | proven | k = 70 has no known prime (k = 50 has exponent 22E7) |
92 | 3, 31 | 32 | proven | proven (k = 15 has exponent 1606, k = 1E has exponent 39260) | ||
95 | 3, 17 | 18 | 7X, since gcd(18+1,95−1) is not 1, and 31 is the next k calculated by the condition, and gcd(31+1,95−1) is not 1, and 65 is the next k calculated by the condition, and gcd(65+1,95−1) is not 1, and 7X is the next k calculated by the condition | proven (k = 4 has exponent 1866, k = 3X has exponent 16E8) | proven | |
96 | 5, 1E | 20 | proven | proven | ||
98 | 3, 11 | 12 | 21, since gcd(12+1,98−1) is not 1, and 21 is the next k calculated by the condition | proven | proven | |
9X | 7, 15 | 42 | 59, since gcd(42+1,9X−1) is not 1, and 59 is the next k calculated by the condition | k = 40 has no known prime | proven | |
9E | 3, 5 | 4 | proven | proven | ||
X2 | 3, 35 | 34 | 12, since 12×X2^{n}−1 has covering set {3, 5, 11} | k = 2X has no known prime | proven | |
X5 | 3, 7 | 8 | proven | proven | ||
X8 | 3, 37 | 38 | k = 34 has no known prime (k = 35 has exponent 1X887, k = 36 has exponent 7635) | proven (k = 1E has exponent 1286, k = 25 has exponent X2274) | ||
X9 | 5, 11 | 12 | proven (k = 6 has exponent 9878) | proven | ||
XE | 3, E | X | proven | proven | ||
E0 | 7, 17 | 18 | 11, since 11×E0^{n}+1 has covering set {5, 7, 15} | proven | proven | |
E2 | 3, 5 | 4 | proven | proven | ||
E5 | 3, 1E | 1X | proven | proven | ||
E7 | 5, 7 | 6 | proven | proven | ||
E8 | 3, 3E | 3X | k = 8 has no known prime (k = 14 has exponent 101436) | proven | ||
EX | E, 11 | 10 | proven | proven | ||
100 | 5, 25 | 4E | proven (k = 2X has exponent 1931) | proven |