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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 βˆ’1 mod q, and all numbers = βˆ’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 11Γ—32nβˆ’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+1,54βˆ’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+1,64βˆ’1) is not 1, and 37 is the next k calculated by the condition X0, since gcd(2Xβˆ’1,64βˆ’1) is not 1, and 37 is the next k calculated by the condition, and gcd(37βˆ’1,64βˆ’1) is not 1, and 93 is the next k calculated by the condition, and gcd(93βˆ’1,64βˆ’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(14βˆ’1,85βˆ’1) is not 1, and 2E is the next k calculated by the condition, and gcd(2Eβˆ’1,85βˆ’1) is not 1, and 57 is the next k calculated by the condition, and gcd(57βˆ’1,85βˆ’1) is not 1, and 72 is the next k calculated by the condition, and gcd(72βˆ’1,85βˆ’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+1,91βˆ’1) is not 1, and 2X is the next k calculated by the condition 100, since gcd(19βˆ’1,91βˆ’1) is not 1, and 2X is the next k calculated by the condition, and gcd(2Xβˆ’1,91βˆ’1) is not 1, and 64 is the next k calculated by the condition, and gcd(64βˆ’1,91βˆ’1) is not 1, and 75 is the next k calculated by the condition, and gcd(75βˆ’1,91βˆ’1) is not 1, and XE is the next k calculated by the condition, and gcd(XEβˆ’1,91βˆ’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+1,95βˆ’1) is not 1, and 31 is the next k calculated by the condition, and gcd(31+1,95βˆ’1) is not 1, and 65 is the next k calculated by the condition, and gcd(65+1,95βˆ’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+1,98βˆ’1) is not 1, and 21 is the next k calculated by the condition proven proven
9X 7, 15 42 59, since gcd(42+1,9Xβˆ’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 12Γ—X2nβˆ’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 11Γ—E0n+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
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