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Williams primes of the form (b-1)•bn+1
Primes table
There are 189 sequences:
| b | Last update | max n | # | Primes for n |
|---|---|---|---|---|
| 2 | 2010-10-01 FermatSearch |
8,589,934,591 | 5 | 1, 2, 4, 8, 16 |
| 3 | 2024-09-14 | 2,000,000 | 48 | 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132, 180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225, 7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106, 152529, 165896, 191814, 529680, 1074726, 1086112, 1175232, 1277862, 1346542, 3123036, 3648969, 5570081, 6236772, 10852677, 11879700 |
| 4 | 2024-10-13 PrimeGrid |
11,010,905 | 29 | 1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, 17175, 21147, 24075, 27396, 27591, 40095, 354984, 400989, 916248, 1145805, 2541153, 5414673, 8204409 |
| 5 | 2024-11-08 | 700,000 | 20 | 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538, 851878, 1052422, 1496566, 5380542, 8431178, 11786358 |
| 6 | 2023-10-27 | 1,000,000 | 29 | 1, 2, 4, 17, 136, 147, 203, 590, 754, 964, 970, 1847, 2031, 2727, 2871, 5442, 7035, 7266, 11230, 23307, 27795, 34152, 42614, 127206, 133086, 231128, 235508, 413426, 880336 |
| 7 | 2014-11-14 | 200,000 | 12 | 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572 |
| 8 | 2022-07-21 | 3,000,000 | 18 | 2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254, 1925332 |
| 9 | 2020-08-08 | 625,000 | 17 | 1, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930, 586240 |
| 10 | 2024-08-26 | 1,000,000 | 26 | 3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, 380734, 583696, 719055, 823037, 862868 |
| 11 | 2024-08-26 Ryan Propper |
200,000 | 7 | 10, 24, 864, 2440, 9438, 68272, 148602 |
| 12 | 2024-08-28 | 200,000 | 9 | 3, 4, 35, 119, 476, 507, 6471, 13319, 31799 |
| 13 | 2024-10-03 | 200,000 | 18 | 1, 2, 4, 21, 34, 48, 53, 160, 198, 417, 773, 1220, 5361, 6138, 15557, 18098, 30580, 75198 |
| 14 | 2024-10-12 | 200,000 | 7 | 2, 40, 402, 1070, 6840, 39006, 75264 |
| 15 | 2024-10-12 | 200,000 | 18 | 1, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, 3692, 3748, 10996, 16504, 151536 |
| 16 | 2024-09-02 | 2,250,000 | 24 | 1, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, 8556, 8565, 10847, 12111, 75122, 183600, 307400, 342107, 416936, 696485, 1061596, 2457527 |
| 17 | 2024-10-24 | 200,000 | 9 | 4, 20, 320, 736, 2388, 3344, 8140, 26908, 111440 |
| 18 | 2024-10-24 | 200,000 | 12 | 1, 6, 9, 12, 22, 30, 102, 154, 600, 33009, 56470, 145053 |
| 19 | 2024-11-04 | 200,000 | 8 | 29, 32, 59, 65, 303, 1697, 5358, 9048 |
| 20 | 2024-09-09 | 100,000 | 8 | 14, 18, 20, 38, 108, 150, 640, 8244 |
| 21 | 2024-09-09 | 100,000 | 22 | 1, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, 3738, 7092, 8756, 15537, 19254, 24712, 35002 |
| 22 | 2024-11-04 | 200,000 | 12 | 1, 9, 53, 261, 1491, 2120, 2592, 6665, 9460, 15412, 24449, 128512 |
| 23 | 2024-09-10 | 100,000 | 9 | 14, 62, 84, 8322, 9396, 10496, 24936, 35874, 47236 |
| 24 | 2024-09-14 | 100,000 | 18 | 2, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, 22272, 38002, 46650 |
| 25 | 2024-11-04 | 200,000 | 9 | 1, 4, 162, 1359, 2620, 26886, 75549, 148606, 183756 |
| 26 | 2024-09-23 | 100,000 | 7 | 2, 18, 100, 1178, 1196, 16644, 81402 |
| 27 | 2024-09-23 | 333,333 | 21 | 4, 5, 167, 408, 416, 701, 707, 1811, 3268, 3508, 7020, 7623, 16449, 35920, 55161, 77645, 87172, 129328, 134853, 217643, 478625 |
| 28 | 2024-11-04 | 200,000 | 13 | 1, 2, 136, 154, 524, 1234, 2150, 2368, 7222, 10082, 14510, 16928, 39400 |
| 29 | 2024-09-25 | 100,000 | 7 | 2, 4, 6, 44, 334, 24714, 41540 |
| 30 | 2018-03-17 | 25,000 | 13 | 4, 5, 9, 18, 71, 124, 165, 172, 888, 2218, 3852, 17871, 23262 |
| 31 | 2018-03-17 | 25,000 | 8 | 5, 18, 93, 269, 488, 1514, 3424, 10858 |
| 32 | 2021-04-01 | 1,800,000 | 4 | 12, 28, 1112164, 1669600 |
| 33 | 2019-06-18 | 10,000 | 12 | 2, 4, 6, 9, 29, 292, 377, 444, 581, 2262, 4533, 7196 |
| 34 | 2019-06-18 | 10,000 | 11 | 1, 2, 4, 9, 32, 33, 50, 56, 168, 315, 1592 |
| 35 | 2019-06-18 | 10,000 | 2 | 2, 6040 |
| 36 | 2020-11-15 | 50,000 | 10 | 2, 9, 11, 21, 50, 96, 137, 210, 345, 1282 |
| 37 | 2020-11-17 | 50,000 | 10 | 9, 16, 36, 72, 299, 532, 681, 3756, 4716, 7936 |
| 38 | 2018-04-01 | 25,000 | 3 | 16, 32, 72 |
| 39 | 2019-06-18 | 10,000 | 5 | 1, 4, 45, 2877, 8314 |
| 40 | 2019-06-18 | 10,000 | 14 | 2, 4, 11, 17, 60, 69, 96, 159, 310, 330, 994, 1103, 7486, 9616 |
| 41 | 2020-08-13 | 50,000 | 3 | 80, 120, 170 |
| 42 | 2019-06-18 | 10,000 | 8 | 1, 4, 5, 15, 20, 39, 3041, 3204 |
| 43 | 2019-06-18 | 10,000 | 9 | 2, 9, 14, 40, 81, 156, 221, 252, 569 |
| 44 | 2019-06-18 | 10,000 | 8 | 4, 108, 174, 272, 1098, 2260, 4972, 7448 |
| 45 | 2019-06-18 | 10,000 | 10 | 2, 5, 15, 17, 28, 84, 390, 778, 1536, 4094 |
| 46 | 2019-06-18 | 10,000 | 13 | 3, 18, 27, 28, 66, 129, 189, 399, 550, 714, 2043, 7350, 7449 |
| 47 | 2019-06-18 | 10,000 | 2 | 16, 4136 |
| 48 | 2019-06-18 | 10,000 | 6 | 2, 5, 6, 40, 986, 1246 |
| 49 | 2020-07-24 | 50,000 | 19 | 2, 4, 9, 11, 26, 63, 75, 135, 164, 449, 662, 1376, 1683, 6644, 15129, 15291, 21761, 35138, 43191 |
| 50 | 2019-06-18 | 10,000 | 6 | 2, 30, 412, 628, 1248, 4190 |
| 51 | 2019-06-18 | 10,000 | 8 | 1, 2, 10, 41, 125, 137, 2204, 9117 |
| 52 | 2019-06-18 | 10,000 | 4 | 15, 28, 372, 1000 |
| 53 | 2019-06-18 | 10,000 | 1 | 960 |
| 54 | 2019-06-18 | 10,000 | 10 | 15, 65, 98, 120, 124, 168, 188, 1371, 1936, 4605 |
| 55 | 2019-06-18 | 10,000 | 9 | 1, 2, 3, 5, 28, 29, 98, 291, 5056 |
| 56 | 2019-06-18 | 10,000 | 7 | 4, 8, 34, 36, 458, 496, 5782 |
| 57 | 2019-06-18 | 10,000 | 6 | 3, 16, 36, 48, 3599, 4383 |
| 58 | 2019-06-18 | 10,000 | 6 | 1, 2, 14, 2514, 4970, 5526 |
| 59 | 2024-10-26 Richard Kapek |
300,000 | 10 | 14, 38, 104, 362, 1256, 3260, 17182, 18166, 41726, 141008 |
| 60 | 2019-06-18 | 10,000 | 10 | 1, 21, 51, 117, 178, 292, 579, 624, 718, 6472 |
| 61 | 2019-06-18 | 10,000 | 7 | 6, 8, 27, 47, 54, 184, 6212 |
| 62 | 2019-06-18 | 10,000 | 2 | 20, 212 |
| 63 | 2019-06-18 | 10,000 | 7 | 1, 4, 17, 18, 484, 1152, 6910 |
| 64 | 2024-07-25 PrimeGrid |
866,666 | 17 | 3, 10, 38, 128, 215, 404, 413, 504, 826, 5679, 70838, 148694, 149718, 255148, 270995, 467855, 1290531 |
| 65 | 2024-10-11 | 100,000 | 3 | 946, 1022, 10898 |
| 66 | 2024-10-11 | 100,000 | 9 | 6, 83, 177, 356, 1468, 5562, 43404, 67500, 89990 |
| 67 | 2024-10-14 | 100,000 | 9 | 1, 3, 4, 112, 2103, 4041, 6567, 15531, 33831 |
| 68 | 2024-10-18 | 100,000 | 7 | 18, 284, 422, 482, 536, 558, 28192 |
| 69 | 2024-10-18 | 100,000 | 13 | 10, 12, 15, 32, 172, 399, 424, 501, 579, 706, 1064, 6314, 13996 |
| 70 | 2024-10-14 | 100,000 | 11 | 1, 11, 16, 40, 479, 1149, 1496, 3279, 17289, 20214, 40779 |
| 71 | 2024-10-24 | 100,000 | 6 | 4, 62, 158, 418, 13106, 87362 |
| 72 | 2024-10-24 | 100,000 | 9 | 1, 3, 5, 12, 15, 140, 292, 15304, 59408 |
| 73 | 2024-09-29 | 30,000 | 5 | 5, 24, 34, 96, 1025 |
| 74 | 2024-10-03 | 25,000 | 2 | 42, 300 |
| 75 | 2024-10-03 | 25,000 | 10 | 4, 6, 9, 95, 516, 628, 830, 1256, 2045, 14805 |
| 76 | 2024-09-29 | 25,000 | 10 | 1, 2, 3, 11, 110, 4644, 6016, 6346, 14106, 21266 |
| 77 | 2024-10-03 | 25,000 | 2 | 828, 5460 |
| 78 | 2024-10-03 | 25,000 | 4 | 1, 5, 116, 9116 |
| 79 | 2024-09-29 | 30,000 | 14 | 1, 10, 12, 24, 101, 124, 288, 300, 804, 857, 1036, 5369, 8020, 27288 |
| 80 | 2024-10-03 | 25,000 | 5 | 2, 4, 346, 5972, 6516 |
| 81 | 2024-10-03 | 50,000 | 3 | 1, 5394, 11811 |
| 82 | 2024-09-29 | 25,000 | 8 | 12, 17, 92, 96, 408, 8687, 11976, 12299 |
| 83 | 2018-04-14 | 25,000 | 6 | 2, 4, 16, 146, 1298, 13330 |
| 84 | 2024-10-03 | 25,000 | 13 | 6, 12, 63, 176, 207, 227, 602, 627, 674, 1516, 10095, 11195, 15572 |
| 85 | 2024-09-29 | 25,000 | 9 | 4, 20, 72, 584, 1394, 5398, 10890, 15084, 19610 |
| 86 | 2024-10-03 | 25,000 | 5 | 30, 158, 4510, 9848, 13610 |
| 87 | 2024-10-03 | 25,000 | 5 | 3, 327, 359, 429, 968 |
| 88 | 2018-04-08 | 25,000 | 3 | 3022, 20556, 23182 |
| 89 | 2024-10-03 | 30,000 | 5 | 2, 110, 214, 302, 314 |
| 90 | 2024-10-01 | 25,000 | 9 | 1, 2, 3, 4, 18, 239, 11003, 11331, 20121 |
| 91 | 2024-09-29 | 25,000 | 8 | 1, 9, 21, 117, 389, 772, 1947, 4274 |
| 92 | 2024-10-01 | 25,000 | 2 | 8, 19888 |
| 93 | 2018-04-08 | 25,000 | 14 | 2, 5, 9, 24, 28, 32, 93, 128, 384, 392, 396, 897, 2464, 23708 |
| 94 | 2024-09-29 | 25,000 | 11 | 4, 8, 27, 41, 564, 744, 1107, 2738, 4876, 17896, 24437 |
| 95 | 2024-10-01 | 25,000 | 5 | 4, 136, 490, 3444, 4320 |
| 96 | 2024-10-01 | 25,000 | 13 | 2, 4, 11, 20, 34, 40, 48, 57, 82, 1713, 3099, 3314, 7209 |
| 97 | 2024-09-29 | 30,000 | 8 | 11, 75, 1023, 1595, 1956, 5516, 9521, 12311 |
| 98 | 2024-10-01 | 25,000 | 2 | 8, 10 |
| 99 | 2024-10-01 | 25,000 | 4 | 2, 3422, 3615, 12162 |
| 100 | 2024-09-29 | 50,000 | 12 | 1, 2, 4, 8, 10, 12, 36, 100, 227, 312, 527, 1030 |
| 101 | 2024-09-29 | 25,000 | 8 | 2, 16, 48, 242, 1182, 2910, 3200, 13666 |
| 102 | 2024-09-29 | 25,000 | 5 | 1, 4, 51, 69, 4011 |
| 103 | 2024-09-29 | 25,000 | 9 | 56, 254, 333, 1936, 2992, 4730, 8093, 9934, 20085 |
| 104 | 2024-09-29 | 25,000 | 4 | 2, 16, 62, 5280 |
| 105 | 2024-09-29 | 25,000 | 11 | 12, 20, 486, 1332, 4182, 4260, 7248, 7443, 22991, 23750, 24978 |
| 106 | 2024-09-29 | 25,000 | 5 | 1, 11, 161, 479, 3053 |
| 107 | 2018-04-08 | 25,000 | 1 | 4 |
| 108 | 2024-09-29 | 25,000 | 3 | 5, 33, 2410 |
| 109 | 2024-09-29 | 25,000 | 10 | 15, 30, 138, 195, 282, 1050, 1101, 3213, 3406, 19245 |
| 110 | 2024-09-29 | 25,000 | 5 | 2, 22, 24, 122, 1680 |
| 113 | 2018-04-08 | 25,000 | 7 | 4, 16, 26, 236, 4912, 9864, 22296 |
| 121 | 2020-11-12 | 15,000 | 10 | 2, 6, 10, 15, 206, 515, 1010, 3354, 4049, 6761 |
| 122 | 2019-06-23 | 10,000 | 1 | 6216 |
| 123 | 2024-05-11 | 877,888 | 1 | 865890 |
| 128 | 2024-07-25 PrimeGrid |
742,857 | 3 | 72, 8722, 51120 |
| 151 | 2020-11-13 | 10,000 | 4 | 1, 4, 63, 8463 |
| 158 | 2018-01-25 | 50,000 | 2 | 1620, 5092 |
| 163 | 2020-11-13 | 10,000 | 5 | 1, 26, 65, 88, 2134 |
| 180 | 2019-06-23 | 10,000 | 2 | 2484, 7989 |
| 202 | 2018-04-27 | 50,000 | 1 | 46773 |
| 241 | 2020-11-12 | 10,000 | 2 | 38, 6392 |
| 249 | 2019-06-23 | 10,000 | 1 | 1851 |
| 251 | 2018-06-13 | 103,000 | 1 | 102978 |
| 256 | 2024-07-25 PrimeGrid |
650,000 | 8 | 4, 9, 42, 130, 580, 33226, 51426, 333556 |
| 257 | 2019-06-23 | 10,000 | 1 | 1344 |
| 269 | 2019-06-23 | 10,000 | 1 | 1436 |
| 272 | 2018-04-30 | 20,000 | 1 | 16680 |
| 297 | 2019-05-19 | 20,000 | 1 | 14313 |
| 298 | 2018-04-23 | 67,000 | 1 | 60670 |
| 326 | 2018-05-15 | 67,000 | 1 | 64756 |
| 328 | 2019-05-21 | 10,000 | 4 | 1626, 2006, 4265, 6144 |
| 342 | 2021-03-13 | 300,000 | 0 | |
| 347 | 2018-05-15 | 78,000 | 1 | 69660 |
| 362 | 2024-08-21 Karsten Bonath |
304,000 | 0 | |
| 363 | 2019-05-24 | 143,000 | 1 | 142876 |
| 419 | 2019-05-19 | 20,000 | 1 | 4338 |
| 422 | 2024-08-21 | 300,000 | 0 | |
| 438 | 2024-09-10 | 200,000 | 1 | 126804 |
| 452 | 2018-06-28 | 72,000 | 1 | 71940 |
| 455 | 2019-05-19 | 20,000 | 1 | 1168 |
| 479 | 2024-08-21 Karsten Bonath |
539,000 | 0 | |
| 487 | 2024-09-06 | 600,000 | 0 | |
| 497 | 2019-05-22 | 10,000 | 1 | 4012 |
| 498 | 2019-06-23 | 10,000 | 1 | 1937 |
| 512 | 2024-02-29 | 500,000 | 0 | |
| 536 | 2019-05-22 | 10,000 | 1 | 9762 |
| 542 | 2024-07-25 | 200,000 | 1 | 156680 |
| 543 | 2018-04-25 | 10,041 | 1 | 10041 |
| 545 | 2019-05-23 | 10,000 | 2 | 1822, 3520 |
| 564 | 2019-05-28 | 50,000 | 1 | 38064 |
| 572 | 2019-05-30 | 10,000 | 1 | 1832 |
| 578 | 2019-05-30 | 10,000 | 1 | 2764 |
| 593 | 2019-05-30 | 10,000 | 2 | 1660, 2272 |
| 599 | 2019-06-23 | 10,000 | 1 | 1018 |
| 602 | 2024-09-26 Lleyton Joseph |
200,000 | 0 | |
| 634 | 2019-08-06 | 86,000 | 1 | 84822 |
| 674 | 2019-05-25 | 10,000 | 1 | 6342 |
| 701 | 2019-05-25 | 10,000 | 1 | 8662 |
| 732 | 2019-06-24 | 10,000 | 1 | 1803 |
| 757 | 2017-05-25 | 100,000 | 0 | |
| 758 | 2019-05-29 | 10,000 | 1 | 3014 |
| 767 | 2019-08-12 | 100,000 | 0 | |
| 779 | 2019-06-24 | 10,000 | 1 | 1014 |
| 782 | 2019-05-31 | 10,000 | 2 | 1200, 4216 |
| 788 | 2019-06-02 | 20,000 | 1 | 13540 |
| 797 | 2019-06-02 | 10,000 | 2 | 5188, 7296 |
| 810 | 2019-06-24 | 10,000 | 2 | 2440, 6795 |
| 817 | 2019-08-20 | 100,000 | 0 | |
| 823 | 2019-06-24 | 10,000 | 3 | 1089, 2361, 4509 |
| 827 | 2019-06-03 | 10,000 | 1 | 8452 |
| 830 | 2019-08-21 | 100,000 | 0 | |
| 836 | 2019-06-06 | 10,000 | 1 | 1048 |
| 839 | 2019-06-06 | 10,000 | 3 | 2172, 3000, 5674 |
| 869 | 2019-06-06 | 20,000 | 2 | 12288, 17086 |
| 872 | 2019-08-05 | 100,000 | 0 | |
| 890 | 2019-06-09 | 37,400 | 1 | 37376 |
| 893 | 2019-08-13 | 100,000 | 0 | |
| 923 | 2019-06-13 | 10,000 | 1 | 4470 |
| 932 | 2019-09-17 | 200,000 | 0 | |
| 950 | 2019-06-11 | 6,000 | 1 | 5956 |
| 953 | 2019-06-20 | 64,000 | 1 | 60994 |
| 974 | 2019-06-14 | 10,000 | 1 | 1622 |
| 992 | 2019-10-07 | 200,000 | 0 | |
| 997 | 2017-07-28 | 100,000 | 0 | |
| 999 | 2024-06-23 | 200,000 | 8 | 3, 99, 546, 554, 1406, 1606, 1893, 14796 |
| 1004 | 2019-06-14 | 30,000 | 1 | 29684 |
| 1005 | 2018-07-21 | 100,000 | 0 | |
| 1007 | 2019-06-28 | 100,000 | 0 | |
| 1022 | 2019-05-19 | 20,000 | 2 | 6064, 14496 |
| 1024 | 2022-07-31 | 364,000 | 13 | 38, 344, 1197, 4364, 10901, 13281, 21693, 30158, 64131, 86848, 117785, 142649, 201257 |
Williams primes
| Main |
| Type MM: (b-1)•bn-1 |
| Type MP: (b-1)•bn+1 |
| Type PM: (b+1)•bn-1 |
| Type PP: (b+1)•bn+1 |
