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Williams primes of the form (b-1)•bn-1
Primes table
There are 169 sequences:
| b | Last update | max n | # | Primes for n |
|---|---|---|---|---|
| 2 | 2024-08-11 GIMPS |
118,392,558 | 51 | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933 |
| 3 | 2024-08-01 | 2,000,000 | 38 | 1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104, 2834778 |
| 4 | 2024-07-31 | 10,925,622 | 39 | 1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859, 8874017, 9462494, 10464378 |
| 5 | 2024-08-30 | 1,000,000 | 20 | 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751 |
| 6 | 2023-11-21 | 1,000,000 | 29 | 1, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706, 386450, 605168, 616879, 1204077 |
| 7 | 2015-11-13 | 200,000 | 21 | 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326 |
| 8 | 2024-06-13 | 4,000,000 | 12 | 3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299, 1743223, 3537579, 4095347 |
| 9 | 2014-03-16 | 200,000 | 17 | 1, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199 |
| 10 | 2024-08-26 | 230,000 | 22 | 1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567, 1762063 |
| 11 | 2024-08-26 | 200,000 | 13 | 1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, 263893 |
| 12 | 2024-08-28 | 200,000 | 16 | 1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961 |
| 13 | 2024-08-28 | 100,000 | 16 | 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466 |
| 14 | 2024-08-30 | 100,000 | 13 | 1, 3, 5, 27, 35, 165, 209, 2351, 11277, 21807, 25453, 52443, 63069 |
| 15 | 2024-08-30 | 100,000 | 5 | 14, 33, 43, 20885, 30323 |
| 16 | 2024-09-02 | 1,750,000 | 10 | 1, 20, 29, 43, 56, 251, 25985, 27031, 142195, 164066 |
| 17 | 2024-09-02 | 100,000 | 9 | 1, 3, 71, 139, 265, 793, 1729, 18069, 80585 |
| 18 | 2024-08-22 Mark Rodenkirch |
25,000 | 10 | 2, 6, 26, 79, 91, 96, 416, 554, 1910, 4968 |
| 19 | 2024-08-22 Mark Rodenkirch |
30,000 | 8 | 6, 9, 20, 43, 174, 273, 428, 1388 |
| 20 | 2024-08-16 Mark Rodenkirch |
25,000 | 4 | 1, 219, 223, 3659 |
| 21 | 2024-08-19 Mark Rodenkirch |
25,000 | 13 | 1, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, 8673 |
| 22 | 2024-08-19 Mark Rodenkirch |
25,000 | 10 | 1, 2, 5, 19, 141, 302, 337, 4746, 5759, 16530 |
| 23 | 2024-08-31 Mark Rodenkirch, Richard Kapek |
25,000 | 7 | 55, 103, 115, 131, 535, 1183, 9683 |
| 24 | 2024-08-19 Mark Rodenkirch |
25,000 | 8 | 12, 18, 63, 153, 221, 1256, 13116, 15593 |
| 25 | 2024-08-19 Mark Rodenkirch |
25,000 | 13 | 1, 5, 7, 30, 75, 371, 383, 609, 819, 855, 7130, 7827, 9368 |
| 26 | 2024-08-19 Mark Rodenkirch |
25,000 | 4 | 133, 205, 215, 1649 |
| 27 | 2024-08-19 Mark Rodenkirch |
25,000 | 1 | 1, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, 6159, 6393, 9013 |
| 28 | 2024-08-19 Mark Rodenkirch |
25,000 | 4 | 20, 1091, 5747, 6770 |
| 29 | 2024-08-19 Mark Rodenkirch |
25,000 | 7 | 1, 7, 11, 57, 69, 235, 16487 |
| 30 | 2018-03-17 | 25,000 | 11 | 2, 83, 566, 938, 1934, 2323, 3032, 7889, 8353, 9899, 11785 |
| 31 | 2018-03-17 | 25,000 | 13 | 1, 5, 27, 41, 48, 177, 398, 536, 608, 618, 13728, 14393, 14949 |
| 32 | 2024-08-27 | 651,837 | 13 | 1, 7, 23, 41, 47, 693, 16331, 30053, 54823, 81221, 133257, 651837, 1817617 |
| 33 | 2019-04-20 | 10,000 | 7 | 2, 3, 146, 1543, 1592, 2247, 3087 |
| 34 | 2019-04-20 | 10,000 | 4 | 15, 25, 99, 188 |
| 35 | 2019-04-20 | 10,000 | 9 | 3, 5, 7, 13, 31, 69, 341, 437, 485 |
| 36 | 2019-04-20 | 10,000 | 9 | 1, 9, 11, 15, 27, 145, 417, 762, 2693 |
| 37 | 2019-04-20 | 10,000 | 5 | 7, 325, 439, 3769, 8537 |
| 38 | 2011-07-12 | 142,000 | 1 | 136211 |
| 39 | 2019-04-20 | 10,000 | 10 | 1, 3, 5, 8, 90, 272, 290, 767, 2775, 6809 |
| 40 | 2019-04-20 | 10,000 | 6 | 1, 19, 218, 767, 2607, 7075 |
| 41 | 2019-04-20 | 10,000 | 3 | 7, 295, 1573 |
| 42 | 2019-04-20 | 10,000 | 12 | 1, 3, 6, 9, 11, 47, 61, 105, 263, 517, 5225, 5783 |
| 43 | 2019-04-20 | 10,000 | 5 | 7, 39, 71, 639, 4590 |
| 44 | 2019-04-20 | 10,000 | 5 | 7, 17, 43, 731, 1751 |
| 45 | 2019-04-20 | 10,000 | 9 | 1, 13, 72, 83, 132, 144, 1473, 1901, 4412 |
| 46 | 2019-04-21 | 10,000 | 12 | 1, 2, 3, 9, 12, 24, 68, 108, 476, 619, 839, 1561 |
| 47 | 2019-04-21 | 10,000 | 7 | 1, 9, 15, 45, 211, 1633, 1933 |
| 48 | 2019-04-21 | 10,000 | 8 | 2, 3, 14, 48, 215, 324, 518, 2439 |
| 49 | 2019-04-21 | 10,000 | 11 | 1, 5, 18, 21, 35, 84, 121, 189, 2610, 4281, 4579 |
| 50 | 2019-04-21 | 10,000 | 7 | 25, 37, 39, 47, 99, 245, 1589 |
| 51 | 2019-04-21 | 10,000 | 5 | 1, 31, 170, 557, 3644 |
| 52 | 2019-04-21 | 10,000 | 10 | 5, 13, 19, 425, 585, 1833, 2130, 3617, 4999, 8447 |
| 53 | 2019-04-21 | 10,000 | 3 | 3, 251, 4567 |
| 54 | 2019-04-21 | 10,000 | 9 | 1, 5, 14, 321, 427, 620, 902, 1461, 1555 |
| 55 | 2019-04-21 | 10,000 | 9 | 1, 15, 44, 71, 98, 147, 225, 242, 4837 |
| 56 | 2019-04-21 | 10,000 | 4 | 1, 25, 77, 297 |
| 57 | 2019-04-21 | 10,000 | 5 | 1, 2, 3, 18, 367 |
| 58 | 2019-04-21 | 10,000 | 3 | 2, 14, 1076 |
| 59 | 2019-04-21 | 10,000 | 4 | 3, 15, 493, 2183 |
| 60 | 2019-04-21 | 10,000 | 3 | 1, 5, 13 |
| 61 | 2019-04-21 | 10,000 | 10 | 1, 2, 7, 25, 132, 216, 439, 482, 581, 1081 |
| 62 | 2019-04-18 | 10,000 | 3 | 899, 2185, 5641 |
| 63 | 2019-04-21 | 10,000 | 8 | 3, 30, 36, 79, 527, 1002, 2687, 8967 |
| 64 | 2021-03-21 | 833,333 | 10 | 11, 21, 119, 1677, 3939, 18759, 19151, 20629, 36449, 101784 |
| 71 | 2018-04-08 | 25,000 | 5 | 1, 59, 93, 2093, 5485 |
| 83 | 2008-08-22 | 29,000 | 1 | 21495 |
| 88 | 2018-04-08 | 25,000 | 5 | 3, 163, 366, 522, 4571 |
| 93 | 2018-04-08 | 25,000 | 2 | 476, 908 |
| 107 | 2018-04-08 | 25,000 | 6 | 37, 103, 319, 583, 605, 5317 |
| 113 | 2012-10-01 | 290,000 | 1 | 286643 |
| 128 | 2024-04-19 | 2,450,000 | 0 | |
| 158 | 2018-04-15 | 25,000 | 4 | 127, 263, 323, 6119 |
| 188 | 2008-07-17 | 20,000 | 1 | 13507 |
| 233 | 2014-10-02 | 250,000 | 0 | |
| 268 | 2022-10-31 | 500,000 | 0 | |
| 293 | 2013-11-07 | 200,000 | 0 | |
| 383 | 2013-09-19 | 200,000 | 0 | |
| 401 | 2013-09-04 | 103,679 | 1 | 103669 |
| 417 | 2011-08-04 | 21,250 | 1 | 21002 |
| 458 | 2011-08-06 | 50,000 | 1 | 46899 |
| 478 | 2013 | 200,000 | 0 | |
| 488 | 2013-12-02 | 200,000 | 0 | |
| 491 | 2006-08-31 | 10,000 | 7 | 1, 5, 9, 75, 109, 2171, 3575 |
| 492 | 2006-08-31 | 10,000 | 5 | 29, 283, 1829, 3283, 4223 |
| 493 | 2006-08-31 | 10,000 | 6 | 2, 15, 67, 2199, 4055, 8103 |
| 494 | 2011-08-09 | 22,000 | 1 | 21579 |
| 495 | 2006-08-31 | 10,000 | 7 | 1, 12, 39, 201, 530, 4434, 6767 |
| 496 | 2006-08-31 | 10,000 | 5 | 1, 6, 186, 281, 517 |
| 497 | 2006-08-31 | 10,000 | 5 | 1, 3, 5, 61, 89 |
| 498 | 2006-08-31 | 10,000 | 4 | 2, 8, 26, 906 |
| 499 | 2006-08-31 | 10,000 | 2 | 32, 67 |
| 500 | 2006-08-31 | 10,000 | 3 | 1, 11, 173 |
| 518 | 2013-09-23 | 131,000 | 1 | 129371 |
| 527 | 2011-08-22 | 67,000 | 1 | 65821 |
| 533 | 2013-12-03 | 200,000 | 0 | |
| 554 | 2014 | 200,000 | 0 | |
| 602 | 2011-08-28 | 22,000 | 1 | 17643 |
| 608 | 2011-08-29 | 37,000 | 1 | 36227 |
| 638 | 2011-09-06 | 76,000 | 1 | 74527 |
| 663 | 2011-09-13 | 75,000 | 1 | 47556 |
| 665 | 2014-05-22 | 200,000 | 0 | |
| 698 | 2011-09-13 | 200,000 | 0 | |
| 758 | 2011-09-17 | 50,563 | 1 | 50563 |
| 779 | 2014-06-11 | 200,000 | 0 | |
| 833 | 2011-09-20 | 16,400 | 1 | 12219 |
| 863 | 2011-10-25 | 200,000 | 0 | |
| 878 | 2014-04-22 | 200,000 | 0 | |
| 918 | 2011-09-12 | 10,000 | 1 | 9756 |
| 932 | 2014-08-20 | 200,000 | 0 | |
| 938 | 2011-10-25 | 51,500 | 1 | 50007 |
| 941 | 2014-08-15 | 200,000 | 0 | |
| 950 | 2011-09-15 | 20,000 | 1 | 16247 |
| 1010 | 2021-05-24 | 300,000 | 0 | |
| 1012 | 2011-11-02 | 33,700 | 1 | 32693 |
| 1034 | 2014-04-08 | 150,000 | 0 | |
| 1088 | 2012-07-09 | 41,000 | 1 | 29579 |
| 1108 | 2012-07-10 | 26,000 | 1 | 15647 |
| 1115 | 2012-07-11 | 4,000 | 1 | 3629 |
| 1142 | 2014-07-11 | 15,000 | 1 | 9501 |
| 1145 | 2012-09-18 | 100,000 | 0 | |
| 1160 | 2012-09-20 | 31,000 | 1 | 24303 |
| 1163 | 2012-09-20 | 5,000 | 1 | 3967 |
| 1164 | 2012-10-31 | 100,000 | 0 | |
| 1199 | 2012-12-03 | 100,000 | 0 | |
| 1253 | 2012-12-20 | 100,000 | 0 | |
| 1268 | 2012-12-21 | 30,000 | 1 | 13735 |
| 1283 | 2024-08-21 | 200,000 | 0 | |
| 1290 | 2019-06-03 | 5,500 | 1 | 5377 |
| 1328 | 2013-01-17 | 100,000 | 0 | |
| 1353 | 2013-02-17 | 100,000 | 0 | |
| 1388 | 2013-02-21 | 52,000 | 1 | 49063 |
| 1430 | 2013-03-25 | 100,000 | 0 | |
| 1444 | 2013-03-25 | 39,000 | 1 | 26345 |
| 1489 | 2013-05-23 | 100,000 | 0 | |
| 1493 | 2013-05-28 | 100,000 | 0 | |
| 1522 | 2013-06-18 | 100,000 | 0 | |
| 1558 | 2013-06 | 11,000 | 2 | 4931, 10944 |
| 1559 | 2013-05-29 | 21,000 | 1 | 20055 |
| 1568 | 2013-05-30 | 17,000 | 1 | 14759 |
| 1632 | 2013-05-30 | 12,200 | 1 | 12169 |
| 1653 | 2013-06-05 | 55,000 | 1 | 21102 |
| 1658 | 2013-06-13 | 100,000 | 0 | |
| 1676 | 2013-06-13 | 34,000 | 1 | 29309 |
| 1683 | 2013-08-16 | 100,000 | 0 | |
| 1696 | 2013-06-18 | 15,700 | 1 | 15488 |
| 1733 | 2013-08-07 | 100,000 | 0 | |
| 1738 | 2013-08-07 | 100,000 | 0 | |
| 1745 | 2013-08-05 | 100,000 | 0 | |
| 1748 | 2013-07-08 | 48,000 | 1 | 24095 |
| 1772 | 2013-07-05 | 33,000 | 1 | 13163 |
| 1793 | 2013-07-08 | 52,000 | 1 | 45859 |
| 1801 | 2019-06-05 | 34,000 | 1 | 23089 |
| 1808 | 2013-07-29 | 100,000 | 0 | |
| 1813 | 2013-07-18 | 46,800 | 1 | 46159 |
| 1820 | 2013-07-18 | 12,000 | 1 | 7051 |
| 1826 | 2013-07-17 | 6,600 | 1 | 6443 |
| 1873 | 2013-08-11 | 61,000 | 1 | 60619 |
| 1893 | 2013-07-24 | 26,000 | 2 | 18264, 21084 |
| 1898 | 2013-07-18 | 5,000 | 1 | 4447 |
| 1901 | 2013-07-18 | 5,700 | 1 | 4177 |
| 1913 | 2013-08-16 | 100,000 | 0 | |
| 1922 | 2013-08-05 | 38,000 | 1 | 36597 |
| 1928 | 2013-08-21 | 100,000 | 0 | |
| 1940 | 2013-08-11 | 62,000 | 1 | 55679 |
| 1948 | 2013-08-29 | 100,000 | 0 | |
| 1967 | 2013-07-24 | 18,800 | 1 | 16993 |
| 1970 | 2013-07-25 | 27,000 | 1 | 13387 |
| 1988 | 2013-09-12 | 100,000 | 0 | |
| 2018 | 2013-09-12 | 200,000 | 0 | |
| 2036 | 2013-07-26 | 13,000 | 1 | 12831 |
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 |
