Williams number

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In number theory, a Williams number base b is a natural number of the form for integers b ≥ 2 and n ≥ 1.[1]The Williams numbers base 2 are exactly the Mersenne numbers.

Williams prime[edit]

A Williams prime is a Williams number that is prime.

Least n ≥ 1 such that (b−1)·bn − 1 is prime are: (start with b = 2)

2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 14, 1, 1, 2, 6, 1, 1, 1, 55, 12, 1, 133, 1, 20, 1, 2, 1, 1, 2, 15, 3, 1, 7, 136211, 1, 1, 7, 1, 7, 7, 1, 1, 1, 2, 1, 25, 1, 5, 3, 1, 1, 1, 1, 2, 3, 1, 1, 899, 3, 11, 1, 1, 1, 63, 1, 13, 1, 25, 8, 3, 2, 7, 1, 44, 2, 11, 3, 81, 21495, 1, 2, 1, 1, 3, 25, 1, 519, 77, 476, 1, 1, 2, 1, 4983, 2, 2, ...
b numbers n ≥ 1 such that (b−1)×bn−1 is prime (these n are checked up to 25000) OEIS sequence
2 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, ... A000043
3 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, ... A003307
4 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, ... A272057
5 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751, ... A046865
6 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, ... A079906
7 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326, ... A046866
8 3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299, ... A268061
9 1, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199, ... A268356
10 1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567, ... A056725
11 1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, 263893, ... A046867
12 1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961, ... A079907
13 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466, ... A297348
14 1, 3, 5, 27, 35, 165, 209, 2351, 11277, 21807, 25453, 52443, ... A273523
15 14, 33, 43, 20885, ...
16 1, 20, 29, 43, 56, 251, 25985, 27031, 142195, 164066, ...
17 1, 3, 71, 139, 265, 793, 1729, 18069, ...
18 2, 6, 26, 79, 91, 96, 416, 554, 1910, 4968, ...
19 6, 9, 20, 43, 174, 273, 428, 1388, ...
20 1, 219, 223, 3659, ...
21 1, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, 8673, ...
22 1, 2, 5, 19, 141, 302, 337, 4746, 5759, 16530, ...
23 55, 103, 115, 131, 535, 1183, 9683, ...
24 12, 18, 63, 153, 221, 1256, 13116, 15593, ...
25 1, 5, 7, 30, 75, 371, 383, 609, 819, 855, 7130, 7827, 9368, ...
26 133, 205, 215, 1649, ...
27 1, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, 6159, 6393, 9013, ...
28 20, 1091, 5747, 6770, ...
29 1, 7, 11, 57, 69, 235, 16487, ...
30 2, 83, 566, 938, 1934, 2323, 3032, 7889, 8353, 9899, 11785, ...

As of September 2018, the largest known Williams prime base 3 is 2×31360104−1.[2]

Generalization[edit]

A Williams number of the second kind base b is a natural number of the form for integers b ≥ 2 and n ≥ 1, a Williams prime of the second kind is a Williams number of the second kind that is prime. The Williams primes of the second kind base 2 are exactly the Fermat primes.

Least n ≥ 1 such that (b−1)·bn + 1 is prime are: (start with b = 2)

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1, 8, 2, 4, 4, 2, 11, 8, 2, 1, ... (sequence A305531 in the OEIS)
b numbers n ≥ 1 such that (b−1)×bn+1 is prime (these n are checked up to 25000) OEIS sequence
2 1, 2, 4, 8, 16, ...
3 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, ... A003306
4 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, ... A326655
5 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538, ... A204322
6 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, ... A247260
7 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572, ... A245241
8 2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254, ... A269544
9 1, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930, ... A056799
10 3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, ... A056797
11 10, 24, 864, 2440, 9438, 68272, 148602, ... A057462
12 3, 4, 35, 119, 476, 507, 6471, 13319, 31799, ... A251259
13 1, 2, 4, 21, 34, 48, 53, 160, 198, 417, 773, 1220, 5361, 6138, 15557, 18098, ...
14 2, 40, 402, 1070, 6840, ...
15 1, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, 3692, 3748, 10996, 16504, ...
16 1, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, 8556, 8565, 10847, 12111, 75122, 183600, 307400, 342107, 416936, ...
17 4, 20, 320, 736, 2388, 3344, 8140, ...
18 1, 6, 9, 12, 22, 30, 102, 154, 600, ...
19 29, 32, 59, 65, 303, 1697, 5358, 9048, ...
20 14, 18, 20, 38, 108, 150, 640, 8244, ...
21 1, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, 3738, 7092, 8756, 15537, 19254, 24712, ...
22 1, 9, 53, 261, 1491, 2120, 2592, 6665, 9460, 15412, 24449, ...
23 14, 62, 84, 8322, 9396, 10496, 24936, ...
24 2, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, 22272, ...
25 1, 4, 162, 1359, 2620, ...
26 2, 18, 100, 1178, 1196, 16644, ...
27 4, 5, 167, 408, 416, 701, 707, 1811, 3268, 3508, 7020, 7623, 16449, ...
28 1, 2, 136, 154, 524, 1234, 2150, 2368, 7222, 10082, 14510, 16928, ...
29 2, 4, 6, 44, 334, 24714, ...
30 4, 5, 9, 18, 71, 124, 165, 172, 888, 2218, 3852, 17871, 23262, ...

As of September 2018, the largest known Williams prime of the second kind base 3 is 2×31175232+1.[3]

A Williams number of the third kind base b is a natural number of the form for integers b ≥ 2 and n ≥ 1, the Williams number of the third kind base 2 are exactly the Thabit numbers. A Williams prime of the third kind is a Williams number of the third kind that is prime.

A Williams number of the fourth kind base b is a natural number of the form for integers b ≥ 2 and n ≥ 1, a Williams prime of the fourth kind is a Williams number of the fourth kind that is prime, such primes do not exist for .

b numbers n such that is prime numbers n such that is prime
2 OEISA002235 OEISA002253
3 OEISA005540 OEISA005537
5 OEISA257790 OEISA143279
10 OEISA111391 (not exist)

It is conjectured that for every b ≥ 2, there are infinitely many Williams primes of the first kind (the original Williams primes) base b, infinitely many Williams primes of the second kind base b, and infinitely many Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many Williams primes of the fourth kind base b.

Dual form[edit]

If we let n take negative values, and choose the numerator of the numbers, then we get these numbers:

Dual Williams numbers of the first kind base b: numbers of the form with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the second kind base b: numbers of the form with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the third kind base b: numbers of the form with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the fourth kind base b: numbers of the form with b ≥ 2 and n ≥ 1. (not exist when b = 1 mod 3)

Unlike the original Williams primes of each kind, some large dual Williams primes of each kind are only probable primes, since for these primes N, neither N−1 not N+1 can be trivially written into a product.

b numbers n such that is (probable) prime (dual Williams primes of the first kind) numbers n such that is (probable) prime (dual Williams primes of the second kind) numbers n such that is (probable) prime (dual Williams primes of the third kind) numbers n such that is (probable) prime (dual Williams primes of the fourth kind)
2 OEISA000043 (see Fermat prime) OEISA050414 OEISA057732
3 OEISA014224 OEISA051783 OEISA058959 OEISA058958
4 OEISA059266 OEISA089437 OEISA217348 (not exist)
5 OEISA059613 OEISA124621 OEISA165701 OEISA089142
6 OEISA059614 OEISA145106 OEISA217352 OEISA217351
7 OEISA191469 OEISA217130 OEISA217131 (not exist)
8 OEISA217380 OEISA217381 OEISA217383 OEISA217382
9 OEISA177093 OEISA217385 OEISA217493 OEISA217492
10 OEISA095714 OEISA088275 OEISA092767 (not exist)

(for the smallest dual Williams primes of the 1st, 2nd and 3rd kinds base b, see OEISA113516, OEISA076845 and OEISA178250)

It is conjectured that for every b ≥ 2, there are infinitely many dual Williams primes of the first kind (the original Williams primes) base b, infinitely many dual Williams primes of the second kind base b, and infinitely many dual Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many dual Williams primes of the fourth kind base b.

See also[edit]

  • Thabit number, which is exactly the Williams number of the third kind base 2

References[edit]

External links[edit]