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Williams primes of the form (b+1)•bn+1
Primes table
There are 110 sequences:
| b | Last update | max n | # | Primes for n |
|---|---|---|---|---|
| 2 | 2024-10-13 PrimeGrid |
22,021,811 | 50 | 1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912, 20909, 34350, 42294, 42665, 44685, 48150, 54792, 55182, 59973, 80190, 157169, 213321, 303093, 362765, 382449, 709968, 801978, 916773, 1832496, 2145353, 2291610, 2478785, 5082306, 7033641, 10829346, 16408818 |
| 3 | 2024-09-10 Ryan Propper |
5,000,000 | 33 | 1, 2, 3, 6, 14, 15, 39, 201, 249, 885, 1005, 1254, 1635, 3306, 3522, 9602, 19785, 72698, 233583, 328689, 537918, 887535, 980925, 1154598, 1499606, 1936890, 2016951, 2143374, 4020126, 6402015, 7578378, 9214845, 11279466 |
| 5 | 2024-09-02 | 10,000,000 | 21 | 1, 2, 3, 23, 27, 33, 63, 158, 278, 290, 351, 471, 797, 8462, 28793, 266030, 439191, 862923, 1392287, 1403337, 6546983 |
| 6 | 2024-09-02 | 7,000,000 | 32 | 1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555, 1128, 1479, 1574, 2808, 3525, 5334, 9980, 13485, 17616, 18846, 24896, 34215, 67914, 115268, 166407, 1246814, 1793775, 2104746, 2396573, 3072198, 6772401 |
| 8 | 2024-07-25 PrimeGrid |
1,733,333 | 27 | 1, 2, 11, 14, 21, 27, 54, 122, 221, 435, 498, 942, 1118, 1139, 1230, 1614, 1934, 13062, 16634, 128330, 278270, 350342, 1165814, 3259421, 3788762, 4135629, 4444829 |
| 9 | 2020-09-13 | 100,000 | 21 | 2, 6, 9, 11, 51, 56, 81, 941, 1647, 7466, 9477, 9806, 11799, 16941, 33437, 35273, 43284, 93813, 95369, 576837, 1775223 |
| 11 | 2024-09-27 | 200,000 | 13 | 2, 3, 6, 8, 138, 149, 222, 363, 995, 1218, 2072, 2559, 93713 |
| 12 | 2024-10-03 | 200,000 | 12 | 1, 2, 8, 9, 17, 26, 62, 86, 152, 365, 2540, 94389 |
| 14 | 2024-10-12 | 200,000 | 15 | 1, 3, 24, 29, 44, 48, 1350, 1520, 4500, 4668, 38553, 86966, 104390, 159783, 187418 |
| 15 | 2024-10-12 | 200,000 | 11 | 1, 3, 5, 21, 35, 101, 860, 2691, 5250, 8787, 9990 |
| 17 | 2024-10-24 | 200,000 | 5 | 1, 6, 204, 2850, 133716 |
| 18 | 2024-10-24 | 200,000 | 6 | 9, 1529, 2477, 4130, 15218, 170613 |
| 20 | 2024-09-09 | 100,000 | 11 | 1, 6, 191, 468, 644, 1583, 3926, 17163, 24371, 57348, 80487 |
| 21 | 2024-09-09 | 100,000 | 8 | 1, 5, 26, 206, 350, 1547, 1853, 7796 |
| 23 | 2024-09-10 | 100,000 | 12 | 2, 11, 18, 27, 57, 362, 383, 651, 3783, 8603, 27726, 43097 |
| 24 | 2024-09-14 | 100,000 | 12 | 1, 2, 3, 8, 66, 968, 4758, 5790, 17168, 30153, 47649, 79478 |
| 26 | 2024-09-23 | 100,000 | 13 | 2, 5, 179, 377, 425, 491, 2108, 2852, 4517, 15836, 34046, 46466, 70790 |
| 27 | 2024-09-23 | 100,000 | 8 | 1, 5, 6, 165, 825, 27798, 39672, 55730 |
| 29 | 2024-09-25 | 100,000 | 11 | 5, 41, 42, 44, 125, 2013, 3398, 4583, 9032, 40461, 92375 |
| 30 | 2018-03-17 | 25,000 | 9 | 2, 36, 282, 425, 918, 2226, 2274, 3354, 4245 |
| 32 | 2021-04-01 | 1,800,000 | 8 | 5, 105, 284, 326, 8537, 9561, 120054, 729962 |
| 33 | 2019-06-25 | 10,000 | 6 | 1, 629, 699, 1571, 1683, 4286 |
| 35 | 2019-06-25 | 10,000 | 8 | 2, 3, 6, 8, 29, 185, 492, 2028 |
| 36 | 2024-10-03 | 25,000 | 5 | 3, 8, 1163, 7562, 12111 |
| 38 | 2018-04-01 | 25,000 | 6 | 1, 21, 63, 87, 122, 206 |
| 39 | 2019-06-25 | 10,000 | 9 | 3, 9, 44, 89, 356, 1208, 1719, 2915, 4139 |
| 41 | 2020-08-13 | 50,000 | 8 | 1, 6, 11, 39, 636, 1637, 12011, 28419 |
| 42 | 2019-06-25 | 10,000 | 7 | 2, 17, 114, 501, 4793, 5189, 5216 |
| 44 | 2019-06-25 | 10,000 | 6 | 2, 3, 24, 26, 32, 134 |
| 45 | 2019-06-25 | 10,000 | 10 | 2, 15, 21, 39, 183, 255, 849, 1604, 1983, 3314 |
| 47 | 2019-06-25 | 10,000 | 4 | 2, 41, 129, 4089 |
| 48 | 2019-06-25 | 10,000 | 7 | 6, 14, 15, 87, 231, 3125, 9773 |
| 50 | 2019-06-25 | 10,000 | 3 | 1, 6, 2987 |
| 51 | 2019-06-25 | 10,000 | 6 | 183, 351, 464, 1175, 1593, 2084 |
| 53 | 2019-06-25 | 10,000 | 7 | 2, 3, 9, 17, 41, 51, 7179 |
| 54 | 2019-06-25 | 10,000 | 11 | 1, 9, 23, 51, 75, 104, 176, 1233, 3243, 7040, 9447 |
| 56 | 2019-06-25 | 10,000 | 9 | 2, 3, 12, 24, 36, 222, 473, 1451, 4856 |
| 57 | 2019-06-25 | 10,000 | 7 | 1, 2, 9, 14, 113, 224, 1545 |
| 59 | 2024-10-26 Richard Kapek |
300,000 | 2 | 1, 219 |
| 60 | 2019-06-25 | 10,000 | 5 | 21, 98, 1421, 3083, 5405 |
| 62 | 2019-06-25 | 10,000 | 7 | 1, 2, 14, 18, 998, 1034, 3458 |
| 63 | 2019-06-25 | 10,000 | 2 | 185, 1751 |
| 65 | 2024-10-11 | 100,000 | 7 | 3, 5, 110, 155, 900, 4235, 21606 |
| 66 | 2024-10-11 | 100,000 | 5 | 1, 2, 18, 1130, 14625 |
| 68 | 2024-10-18 | 100,000 | 3 | 2, 41891, 76523 |
| 69 | 2024-10-18 | 100,000 | 12 | 1, 2, 5, 9, 113, 429, 2360, 5621, 5976, 6329, 7631, 58311 |
| 71 | 2024-10-24 | 100,000 | 9 | 1, 2, 3, 20, 93, 633, 1035, 1695, 5511 |
| 72 | 2024-10-24 | 100,000 | 4 | 120, 170, 2970, 4970 |
| 74 | 2024-10-03 | 25,000 | 7 | 2, 12, 78, 272, 798, 2415, 14027 |
| 75 | 2024-10-03 | 25,000 | 2 | 1, 1637 |
| 77 | 2024-10-03 | 25,000 | 1 | 1 |
| 78 | 2024-10-03 | 25,000 | 5 | 1, 5, 15, 42, 7733 |
| 80 | 2024-10-03 | 25,000 | 9 | 1, 24, 33, 147, 222, 225, 426, 1575, 3462 |
| 81 | 2024-10-03 | 25,000 | 10 | 8, 45, 252, 282, 746, 3248, 6078, 8352, 11012, 15494 |
| 83 | 2018-04-14 | 25,000 | 8 | 5, 21, 198, 803, 1334, 4143, 5666, 14774 |
| 84 | 2024-10-03 | 25,000 | 6 | 9, 257, 348, 429, 662, 10119 |
| 86 | 2024-10-03 | 25,000 | 3 | 2, 2921, 3437 |
| 87 | 2024-10-03 | 25,000 | 9 | 2, 9, 24, 276, 366, 390, 3030, 4266, 6056 |
| 89 | 2024-10-03 | 25,000 | 10 | 1, 2, 3, 8, 230, 873, 927, 1029, 5043, 7319 |
| 90 | 2024-10-01 | 25,000 | 9 | 1, 23, 54, 158, 800, 1143, 2342, 5735, 11652 |
| 92 | 2024-10-01 | 25,000 | 4 | 2, 9, 566, 1677 |
| 93 | 2018-04-08 | 25,000 | 8 | 3, 6, 185, 210, 282, 605, 13961, 17553 |
| 95 | 2024-10-01 | 25,000 | 7 | 9, 339, 515, 567, 10962, 13647, 20420 |
| 96 | 2024-10-01 | 25,000 | 5 | 14, 17, 122, 689, 9560 |
| 98 | 2024-10-01 | 25,000 | 5 | 3, 11, 114, 1083, 1418 |
| 99 | 2024-10-01 | 25,000 | 3 | 1, 11, 30 |
| 101 | 2024-09-29 | 25,000 | 7 | 1, 2, 45, 116, 321, 2271, 13766 |
| 102 | 2024-09-29 | 25,000 | 9 | 5, 14, 113, 194, 314, 921, 5850, 12854, 22136 |
| 104 | 2024-09-29 | 25,000 | 6 | 5, 12, 77, 252, 3411, 5081 |
| 105 | 2024-09-29 | 25,000 | 7 | 1, 23, 485, 878, 1433, 3075, 5235 |
| 107 | 2018-04-08 | 25,000 | 4 | 80, 1568, 2630, 6281 |
| 108 | 2024-09-29 | 25,000 | 3 | 398, 1578, 12926 |
| 110 | 2024-09-29 | 25,000 | 5 | 1, 3, 21, 69, 1664 |
| 113 | 2018-04-08 | 25,000 | 6 | 3, 5, 23, 35, 74, 9753 |
| 128 | 2024-07-25 PrimeGrid |
742,857 | 8 | 3, 41, 138, 153, 8198, 10838, 11718, 16071 |
| 158 | 2018-01-25 | 25,000 | 3 | 141, 225, 5190 |
| 171 | 2019-06-26 | 10,000 | 2 | 1851, 3411 |
| 173 | 2020-06-14 | 10,000 | 9 | 1, 5, 21, 35, 41, 327, 515, 1157, 8918 |
| 179 | 2020-06-14 | 10,000 | 9 | 2, 30, 33, 83, 585, 1970, 4368, 5205, 6681 |
| 201 | 2019-06-26 | 50,000 | 1 | 31274 |
| 222 | 2018-04-27 | 54,000 | 1 | 52725 |
| 227 | 2018-06-19 | 37,000 | 1 | 36321 |
| 269 | 2018-06-19 | 10,000 | 1 | 1398 |
| 275 | 2018-06-19 | 10,000 | 2 | 1911, 8943 |
| 302 | 2018-06-19 | 10,000 | 1 | 1085 |
| 327 | 2020-07-19 | 135,981 | 1 | 135981 |
| 339 | 2019-06-25 | 10,000 | 2 | 1002, 5798 |
| 425 | 2019-05-14 | 11,229 | 1 | 11229 |
| 473 | 2019-05-15 | 20,000 | 2 | 3521, 4031 |
| 512 | 2024-02-29 | 500,000 | 9 | 1, 5, 14, 17, 26, 15164, 21468, 37460, 90996 |
| 542 | 2019-05-15 | 20,000 | 1 | 1806 |
| 633 | 2019-06-25 | 10,000 | 1 | 5246 |
| 683 | 2019-05-15 | 10,000 | 1 | 6774 |
| 710 | 2019-05-20 | 25,000 | 1 | 24110 |
| 717 | 2019-05-21 | 46,000 | 1 | 37506 |
| 719 | 2019-05-21 | 20,000 | 1 | 13418 |
| 813 | 2020-01-06 | 100,000 | 0 | |
| 858 | 2019-06-25 | 10,000 | 1 | 3749 |
| 863 | 2024-08-21 | 200,000 | 0 | |
| 888 | 2018-06-23 | 10,000 | 2 | 1647, 4118 |
| 912 | 2019-06-15 | 10,000 | 1 | 4917 |
| 929 | 2019-06-15 | 10,000 | 2 | 3615, 6446 |
| 947 | 2019-06-17 | 10,000 | 2 | 2693, 8870 |
| 962 | 2020-01-10 | 200,000 | 0 | |
| 983 | 2019-06-17 | 11,000 | 3 | 1875, 7482, 10005 |
| 999 | 2019-06-25 | 10,000 | 1 | 3761 |
| 1008 | 2019-06-17 | 10,000 | 1 | 2771 |
| 1010 | 2019-06-17 | 10,000 | 1 | 3351 |
| 1017 | 2024-01-13 | 350,000 | 0 | |
| 2024 | 2024-10-18 | 20,000 | 5 | 350, 465, 3822, 6294, 10922 |
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 |
