# Williams number

In number theory , a Williams number with base b is a natural number of the form

${\ displaystyle (b-1) \ cdot b ^ {n} -1}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$

They were named after the Canadian mathematician Hugh C. Williams , who first studied these numbers in 1981.

Williams base numbers have the form and are exactly the Mersenne numbers . ${\ displaystyle b = 2}$${\ displaystyle (2-1) \ cdot 2 ^ {n} -1 = 2 ^ {n} -1}$

## Williams primes

A Williams prime is a Williams number that is prime . The smallest , so that it is a prime number, are the following (starting with ): ${\ displaystyle n \ geq 1}$${\ displaystyle (b-1) \ cdot b ^ {n} -1 \ in \ mathbb {P}}$${\ displaystyle 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 ...

It is believed that there are infinitely many Williams primes for the base . ${\ displaystyle b}$

Below is a table which is the smallest Williams primes to the base with be seen (if also solution would be is this because in brackets is actually not allowed, but the completeness led to): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 30}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle (b-1) \ cdot b ^ {n} -1}$ ${\ displaystyle n \ geq 1}$such that Williams are prime numbers ${\ displaystyle (b-1) \ cdot b ^ {n} -1}$ OEIS link
02 ${\ displaystyle 2 ^ {n} -1}$ 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, 371516667, 4261743801, 43201, 7723288516609, 43201121, 7723288516609, 57232912160 82589933 ... (all Mersenne prime exponents) (Follow A000043 in OEIS )
03 ${\ displaystyle 2 \ cdot 3 ^ {n} -1}$ 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 ... (Follow A003307 in OEIS )
04th ${\ displaystyle 3 \ cdot 4 ^ {n} -1}$ (0), 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 ... (Follow A272057 in OEIS )
05 ${\ displaystyle 4 \ cdot 5 ^ {n} -1}$ (0), 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751 ... (Follow A046865 in OEIS )
06th ${\ displaystyle 5 \ cdot 6 ^ {n} -1}$ 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 ... (Follow A079906 in OEIS )
07th ${\ displaystyle 6 \ cdot 7 ^ {n} -1}$ (0), 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326 ... (Follow A046866 in OEIS )
08th ${\ displaystyle 7 \ cdot 8 ^ {n} -1}$ 3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299 ... (Follow A268061 in OEIS )
09 ${\ displaystyle 8 \ cdot 9 ^ {n} -1}$ (0), 1, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199 ... (Follow A268356 in OEIS )
10 ${\ displaystyle 9 \ cdot 10 ^ {n} -1}$ 1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567 ... (Follow A056725 in OEIS )
11 ${\ displaystyle 10 \ cdot 11 ^ {n} -1}$ 1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, 263893 ... (Follow A046867 in OEIS )
12 ${\ displaystyle 11 \ cdot 12 ^ {n} -1}$ 1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961 ... (Follow A079907 in OEIS )
13 ${\ displaystyle 12 \ cdot 13 ^ {n} -1}$ (0), 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466 ... (Follow A297348 in OEIS )
14th ${\ displaystyle 13 \ cdot 14 ^ {n} -1}$ 1, 3, 5, 27, 35, 165, 209, 2351, 11277, 21807, 25453, 52443 ... (Follow A273523 in OEIS )
15th ${\ displaystyle 14 \ cdot 15 ^ {n} -1}$ (0), 14, 33, 43, 20885…
16 ${\ displaystyle 15 \ cdot 16 ^ {n} -1}$ 1, 20, 29, 43, 56, 251, 25985, 27031, 142195, 164066 ...
17th ${\ displaystyle 16 \ cdot 17 ^ {n} -1}$ 1, 3, 71, 139, 265, 793, 1729, 18069 ...
18th ${\ displaystyle 17 \ cdot 18 ^ {n} -1}$ 2, 6, 26, 79, 91, 96, 416, 554, 1910, 4968 ...
19th ${\ displaystyle 18 \ cdot 19 ^ {n} -1}$ (0), 6, 9, 20, 43, 174, 273, 428, 1388 ...
20th ${\ displaystyle 19 \ cdot 20 ^ {n} -1}$ 1, 219, 223, 3659 ...
21st ${\ displaystyle 20 \ cdot 21 ^ {n} -1}$ (0), 1, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, 8673 ...
22nd ${\ displaystyle 21 \ cdot 22 ^ {n} -1}$ 1, 2, 5, 19, 141, 302, 337, 4746, 5759, 16530 ...
23 ${\ displaystyle 22 \ cdot 23 ^ {n} -1}$ 55, 103, 115, 131, 535, 1183, 9683 ...
24 ${\ displaystyle 23 \ cdot 24 ^ {n} -1}$ 12, 18, 63, 153, 221, 1256, 13116, 15593 ...
25th ${\ displaystyle 24 \ cdot 25 ^ {n} -1}$ (0), 1, 5, 7, 30, 75, 371, 383, 609, 819, 855, 7130, 7827, 9368 ...
26th ${\ displaystyle 25 \ cdot 26 ^ {n} -1}$ 133, 205, 215, 1649 ...
27 ${\ displaystyle 26 \ cdot 27 ^ {n} -1}$ 1, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, 6159, 6393, 9013 ...
28 ${\ displaystyle 27 \ cdot 28 ^ {n} -1}$ 20, 1091, 5747, 6770 ...
29 ${\ displaystyle 28 \ cdot 29 ^ {n} -1}$ 1, 7, 11, 57, 69, 235, 16487 ...
30th ${\ displaystyle 29 \ cdot 30 ^ {n} -1}$ 2, 83, 566, 938, 1934, 2323, 3032, 7889, 8353, 9899, ​​11785 ...

The largest currently known Williams prime is at the same time the largest Mersenne number and the largest currently known prime . It was discovered by the American Patrick Laroche on December 21, 2018 and has 24,862,048 positions. (As of January 28, 2020) ${\ displaystyle p = 2 ^ {82589933} -1}$

The largest single-base Williams prime currently known is . It was discovered on June 23, 2015 by Michael Schulz from Germany and has 3,580,969 jobs. (As of January 28, 2020) ${\ displaystyle b \ not = 2}$${\ displaystyle p = 3 \ cdot 4 ^ {5947859} -1 = 3 \ cdot 2 ^ {11895718} -1}$

The largest currently known Williams-prime with a base , is . It was discovered by Borys Jaworski on July 31, 2015 and has 648,935 positions. (As of January 28, 2020) ${\ displaystyle b \ not = 2 ^ {k}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle p = 2 \ cdot 3 ^ {1360104} -1}$

## Generalizations

### Williams numbers of the 2nd kind

A Williams number of the 2nd kind with base b is a natural number of the form

${\ displaystyle (b-1) \ cdot b ^ {n} +1}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$

Williams numbers of the 2nd type for the base have the form and are exactly the Fermat numbers . ${\ displaystyle b = 2}$${\ displaystyle (2-1) \ cdot 2 ^ {n} + 1 = 2 ^ {n} +1}$

A 2nd kind Williams prime is a 2nd kind Williams number, which is prime. The smallest , so that it is a prime number, are the following (starting with ): ${\ displaystyle n \ geq 1}$${\ displaystyle (b-1) \ cdot b ^ {n} +1 \ in \ mathbb {P}}$${\ displaystyle 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 ... (episode A305531 in OEIS )

It is assumed that there are infinitely many Williams primes of the 2nd kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest Williams primes second nature to the base with take may (if also solution would be is it in parenthesis, as is stated but is actually not allowed, completeness with): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 30}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle (b-1) \ cdot b ^ {n} +1}$ ${\ displaystyle n \ geq 1}$so that Williams are primes of the 2nd kind ${\ displaystyle (b-1) \ cdot b ^ {n} +1}$ OEIS link
02 ${\ displaystyle 2 ^ {n} +1}$ 1, 2, 4, 8, 16 ... (all Fermat prime number exponents)
03 ${\ displaystyle 2 \ cdot 3 ^ {n} +1}$ (0), 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 ... (Follow A003306 in OEIS )
04th ${\ displaystyle 3 \ cdot 4 ^ {n} +1}$ 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 ... (Follow A326655 in OEIS )
05 ${\ displaystyle 4 \ cdot 5 ^ {n} +1}$ (0), 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538 ... (Follow A204322 in OEIS )
06th ${\ displaystyle 5 \ cdot 6 ^ {n} +1}$ 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 ... (Follow A247260 in OEIS )
07th ${\ displaystyle 6 \ cdot 7 ^ {n} +1}$ (0), 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572 ... (Follow A245241 in OEIS )
08th ${\ displaystyle 7 \ cdot 8 ^ {n} +1}$ 2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254 ... (Follow A269544 in OEIS )
09 ${\ displaystyle 8 \ cdot 9 ^ {n} +1}$ 1, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930 ... (Follow A056799 in OEIS )
10 ${\ displaystyle 9 \ cdot 10 ^ {n} +1}$ 3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240 ... (Follow A056797 in OEIS )
11 ${\ displaystyle 10 \ cdot 11 ^ {n} +1}$ (0), 10, 24, 864, 2440, 9438, 68272, 148602 ... (Follow A057462 in OEIS )
12 ${\ displaystyle 11 \ cdot 12 ^ {n} +1}$ 3, 4, 35, 119, 476, 507, 6471, 13319, 31799 ... (Follow A251259 in OEIS )
13 ${\ displaystyle 12 \ cdot 13 ^ {n} +1}$ (0), 1, 2, 4, 21, 34, 48, 53, 160, 198, 417, 773, 1220, 5361, 6138, 15557, 18098 ...
14th ${\ displaystyle 13 \ cdot 14 ^ {n} +1}$ 2, 40, 402, 1070, 6840 ...
15th ${\ displaystyle 14 \ cdot 15 ^ {n} +1}$ 1, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, 3692, 3748, 10996, 16504 ...
16 ${\ displaystyle 15 \ cdot 16 ^ {n} +1}$ 1, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, 8556, 8565, 10847, 12111, 75122, 183600, 307400, 342107, 416936 ...
17th ${\ displaystyle 16 \ cdot 17 ^ {n} +1}$ (0), 4, 20, 320, 736, 2388, 3344, 8140 ...
18th ${\ displaystyle 17 \ cdot 18 ^ {n} +1}$ 1, 6, 9, 12, 22, 30, 102, 154, 600 ...
19th ${\ displaystyle 18 \ cdot 19 ^ {n} +1}$ (0), 29, 32, 59, 65, 303, 1697, 5358, 9048 ...
20th ${\ displaystyle 19 \ cdot 20 ^ {n} +1}$ 14, 18, 20, 38, 108, 150, 640, 8244 ...
21st ${\ displaystyle 20 \ cdot 21 ^ {n} +1}$ 1, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, 3738, 7092, 8756, 15537, 19254, 24712 ...
22nd ${\ displaystyle 21 \ cdot 22 ^ {n} +1}$ 1, 9, 53, 261, 1491, 2120, 2592, 6665, 9460, 15412, 24449 ...
23 ${\ displaystyle 22 \ cdot 23 ^ {n} +1}$ (0), 14, 62, 84, 8322, 9396, 10496, 24936 ...
24 ${\ displaystyle 23 \ cdot 24 ^ {n} +1}$ 2, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, 22272 ...
25th ${\ displaystyle 24 \ cdot 25 ^ {n} +1}$ 1, 4, 162, 1359, 2620 ...
26th ${\ displaystyle 25 \ cdot 26 ^ {n} +1}$ 2, 18, 100, 1178, 1196, 16644 ...
27 ${\ displaystyle 26 \ cdot 27 ^ {n} +1}$ 4, 5, 167, 408, 416, 701, 707, 1811, 3268, 3508, 7020, 7623, 16449 ...
28 ${\ displaystyle 27 \ cdot 28 ^ {n} +1}$ 1, 2, 136, 154, 524, 1234, 2150, 2368, 7222, 10082, 14510, 16928 ...
29 ${\ displaystyle 28 \ cdot 29 ^ {n} +1}$ (0), 2, 4, 6, 44, 334, 24714 ...
30th ${\ displaystyle 29 \ cdot 30 ^ {n} +1}$ 4, 5, 9, 18, 71, 124, 165, 172, 888, 2218, 3852, 17871, 23262 ...

The largest currently known Williams prime 2nd species, with a base , is . It was discovered on January 14, 2014 by Sai Yik Tang from Malaysia and has 3,259,959 positions. (As of January 28, 2020) ${\ displaystyle b = 2 ^ {k}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle p = 3 \ cdot 4 ^ {5414673} + 1 = 3 \ cdot 2 ^ {10829346} +1}$

The largest currently known Williams prime 2nd species, with a base , is . It was discovered by David Broadhurst on February 22, 2010 and has 560,729 positions. (As of January 28, 2020) ${\ displaystyle b \ not = 2 ^ {k}}$${\ displaystyle k \ in \ mathbb {N}}$${\ displaystyle p = 2 \ cdot 3 ^ {1175232} +1}$

### Williams numbers of the 3rd kind

A Williams number of the 3rd kind with base b is a natural number of the form

${\ displaystyle (b + 1) \ cdot b ^ {n} -1}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$

It is also called a Thabit number with base b .

Williams numbers of the 3rd kind to the base have the form and are exactly the Thabit numbers . ${\ displaystyle b = 2}$${\ displaystyle (2 + 1) \ cdot 2 ^ {n} -1 = 3 \ cdot 2 ^ {n} -1}$

A type 3 Williams prime is a type 3 Williams number, which is prime.

It is assumed that there are infinitely many Williams primes of the 3rd kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest Williams primes 3. Type the base with take may (if also solution would be is it in parenthesis, as is stated but is actually not allowed, completeness with): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 12}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle (b + 1) \ cdot b ^ {n} -1}$ ${\ displaystyle n \ geq 1}$so that Williams primes of the 3rd kind are ${\ displaystyle (b + 1) \ cdot b ^ {n} -1}$ OEIS link
02 ${\ displaystyle 3 \ cdot 2 ^ {n} -1}$ (0), 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, 827, 1274, 3276, 4204, 5134, 7559, 12676, 14898, 18123, 18819, 25690, 26459, 41628, 51387, 71783, 80330, 85687, 88171, 97063, 123630, 155930, 164987, 234760, 414840, 584995, 702038, 727699, 992700, 1201046, 1232255, 2312734, 3136255, 4235414, 6090515, 11484018, 11731850, 11895718 ... (Follow A002235 in OEIS )
03 ${\ displaystyle 4 \ cdot 3 ^ {n} -1}$ (0), 1, 3, 5, 7, 15, 45, 95, 235, 463, 733, 1437, 1583, 1677, 1803, 4163, 4765, 9219, 9959, 25477, 26059, 41539, 54195, 65057, 74977, 116589, 192289, 311835, 350767, 353635, 416337, 423253 ... (Follow A005540 in OEIS )
04th ${\ displaystyle 5 \ cdot 4 ^ {n} -1}$ 1, 2, 4, 5, 6, 7, 9, 16, 24, 27, 36, 74, 92, 124, 135, 137, 210, 670, 719, 761, 819, 877, 942, 1007, 1085, 1274, 1311, 1326, 1352, 6755 ...
05 ${\ displaystyle 6 \ cdot 5 ^ {n} -1}$ (0), 1, 2, 5, 11, 28, 65, 72, 361, 479, 494, 599, 1062, 1094, 1193, 2827, 3271, 3388, 3990, 4418, 11178, 16294, 25176, 42500, 68320, 85698, 145259, 159119, 169771 ... (Follow A257790 in OEIS )
06th ${\ displaystyle 7 \ cdot 6 ^ {n} -1}$ 1, 2, 3, 13, 21, 28, 30, 32, 36, 48, 52, 76, 734, 2236, 2272, 3135, 3968, 6654, 7059 ...
07th ${\ displaystyle 8 \ cdot 7 ^ {n} -1}$ (0), 4, 7, 10, 14, 23, 59, 1550, 1835, 2515, 3532, 3818, 8260 ...
08th ${\ displaystyle 9 \ cdot 8 ^ {n} -1}$ 1, 5, 7, 21, 33, 53, 103, 313, 517, 1863, 2669, 3849, 4165 ...
09 ${\ displaystyle 10 \ cdot 9 ^ {n} -1}$ 1, 2, 4, 5, 7, 10, 11, 13, 15, 19, 27, 29, 35, 42, 51, 70, 112, 164, 179, 180, 242, 454, 621, 2312, 3553, 6565…
10 ${\ displaystyle 11 \ cdot 10 ^ {n} -1}$ 1, 9, 11, 17, 22, 29, 36, 37, 52, 166, 448, 2011, 3489, 4871, 6982, 10024, 16974, 33287, 47364, 58873, 126160 ... (Follow A111391 in OEIS )
11 ${\ displaystyle 12 \ cdot 11 ^ {n} -1}$ (0), 1, 2, 3, 4, 11, 13, 22, 27, 48, 51, 103, 147, 280, 908, 1346, 1524, 1776, 2173, 2788, 6146 ...
12 ${\ displaystyle 13 \ cdot 12 ^ {n} -1}$ 2, 6, 11, 66, 196, 478, 2968, 3568, 5411, 7790 ...

### Williams numbers of the 4th kind

A Williams number of the 4th kind with base b is a natural number of the form

${\ displaystyle (b + 1) \ cdot b ^ {n} +1}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$

It is also called a Thabit number of the 2nd kind with base b .

Williams numbers of the 4th kind at the base have the form and are exactly the Thabit numbers of the 2nd kind . ${\ displaystyle b = 2}$${\ displaystyle (2 + 1) \ cdot 2 ^ {n} + 1 = 3 \ cdot 2 ^ {n} +1}$

A type 4 Williams prime is a type 4 Williams number, which is prime.

The following applies: There is no Williams prime number of the 4th kind with a base . ${\ displaystyle b \ equiv 1 {\ pmod {3}}}$

Proof:
If is, also applies . Furthermore is . Thus one obtains . So in this case what was to be shown is always divisible by and thus never a prime number.${\ displaystyle b \ equiv 1 {\ pmod {3}}}$${\ displaystyle b ^ {n} \ equiv 1 ^ {n} = 1 {\ pmod {3}}}$${\ displaystyle b + 1 \ equiv 1 + 1 = 2 {\ pmod {3}}}$${\ displaystyle (b + 1) \ cdot b ^ {n} +1 \ equiv 2 \ cdot 1 + 1 = 3 \ equiv 0 {\ pmod {3}}}$${\ displaystyle (b + 1) \ cdot b ^ {n} +1}$${\ displaystyle 3}$${\ displaystyle \ Box}$

It is assumed that there are infinitely many Williams primes of the 4th kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest Williams primes 4. Type the base with take may (if also solution would be is it in parenthesis, as is stated but is actually not allowed, completeness with): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 12}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle (b + 1) \ cdot b ^ {n} +1}$ ${\ displaystyle n \ geq 1}$such that Williams primes are of the 4th kind ${\ displaystyle (b + 1) \ cdot b ^ {n} +1}$ OEIS link
02 ${\ displaystyle 3 \ cdot 2 ^ {n} +1}$ 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, 247872985, 5082306, 7033641… (Follow A002253 in OEIS )
03 ${\ displaystyle 4 \ cdot 3 ^ {n} +1}$ (0), 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… (Follow A005537 in OEIS )
04th ${\ displaystyle 5 \ cdot 4 ^ {n} +1}$ there are no prime numbers because of this form ${\ displaystyle b = 4 \ equiv 1 {\ pmod {3}}}$
05 ${\ displaystyle 6 \ cdot 5 ^ {n} +1}$ (0), 1, 2, 3, 23, 27, 33, 63, 158, 278, 290, 351, 471, 797, 8462, 28793, 266030 ... (Follow A143279 in OEIS )
06th ${\ displaystyle 7 \ cdot 6 ^ {n} +1}$ 1, 6, 17, 38, 50, 80, 207, 236, 264, 309, 555, 1128, 1479, 1574, 2808, 3525, 5334, 9980 ...
07th ${\ displaystyle 8 \ cdot 7 ^ {n} +1}$ there are no prime numbers because of this form ${\ displaystyle b = 7 \ equiv 1 {\ pmod {3}}}$
08th ${\ displaystyle 9 \ cdot 8 ^ {n} +1}$ 1, 2, 11, 14, 21, 27, 54, 122, 221, 435, 498, 942, 1118, 1139, 1230, 1614, 1934 ...
09 ${\ displaystyle 10 \ cdot 9 ^ {n} +1}$ (0), 2, 6, 9, 11, 51, 56, 81, 941, 1647, 7466, 9477, 9806 ...
10 ${\ displaystyle 11 \ cdot 10 ^ {n} +1}$ there are no prime numbers because of this form ${\ displaystyle b = 10 \ equiv 1 {\ pmod {3}}}$
11 ${\ displaystyle 12 \ cdot 11 ^ {n} +1}$ (0), 2, 3, 6, 8, 138, 149, 222, 363, 995, 1218, 2072, 2559 ...
12 ${\ displaystyle 13 \ cdot 12 ^ {n} +1}$ 1, 2, 8, 9, 17, 26, 62, 86, 152, 365, 2540 ...

## Dual Williams numbers

Williams numbers have the form with integer and . But what happens if you let the exponent become negative? So be with us . Then we get: ${\ displaystyle (b \ pm 1) \ cdot b ^ {n} \ pm 1}$${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$${\ displaystyle n}$${\ displaystyle m \ in \ mathbb {Z} ^ {-}}$${\ displaystyle m: = - n}$

${\ displaystyle (b \ pm 1) \ cdot b ^ {m} \ pm 1 = (b \ pm 1) \ cdot b ^ {- n} \ pm 1 = (b \ pm 1) \ cdot {\ frac { 1} {b ^ {n}}} \ pm 1 = {\ frac {(b \ pm 1)} {b ^ {n}}} \ pm {\ frac {b ^ {n}} {b ^ {n }}} = {\ frac {(b \ pm 1) \ pm b ^ {n}} {b ^ {n}}} = {\ frac {\ pm (b ^ {n} \ pm (b \ pm 1 ))} {b ^ {n}}}}$

If you only take the amount of the numerator of this fraction, you get the number . This leads to four new definitions: ${\ displaystyle | b ^ {n} \ pm (b \ pm 1) |}$

• A dual Williams number of the first kind to the base${\ displaystyle b}$ is a natural number of the form
${\ displaystyle b ^ {n} - (b-1)}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$
• A dual base Williams number of the 2nd kind${\ displaystyle b}$ is a natural number of the form
${\ displaystyle b ^ {n} + (b-1)}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$
• A dual base Williams number of the 3rd kind${\ displaystyle b}$ is a natural number of the form
${\ displaystyle b ^ {n} - (b + 1)}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$
• A dual Williams number of the 4th kind to the base${\ displaystyle b}$ is a natural number of the form
${\ displaystyle b ^ {n} + (b + 1)}$with integer and .${\ displaystyle b \ geq 2}$${\ displaystyle n \ geq 1}$

A Williams dual prime number the . Kind to the base${\ displaystyle k}$${\ displaystyle b}$ is a Williams number . Kind to the base , which is prime ( ). ${\ displaystyle k}$${\ displaystyle b}$${\ displaystyle k \ in \ {1,2,3,4 \}}$

In contrast to the Williams primes (of whatever type), there are no primal tests specially tailored to these numbers for the dual Williams primes. Therefore, larger dual Williams primes are often "just" PRP numbers ( probable primes ) because they are too large for one to be able to determine within a reasonable time with known prime number tests whether they are actually prime numbers or perhaps just pseudoprime numbers . This is mainly due to the fact that with the dual Williams prime numbers one can neither write nor simply write as a product (see Lucas test ). ${\ displaystyle N}$${\ displaystyle N-1}$${\ displaystyle N + 1}$

### Dual Williams numbers of the first kind

The smallest , so that it is a prime number, are the following (starting with ): ${\ displaystyle n \ geq 1}$${\ displaystyle b ^ {n} - (b-1) \ in \ mathbb {P}}$${\ displaystyle b = 2}$

2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3 ... ( continuation A113516 in OEIS )

It is assumed that there are an infinite number of dual Williams primes of the 1st kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest dual Williams primes (or -PRP numbers) to the base 1. Species with refer can: ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 10}$

${\ displaystyle b}$ ${\ displaystyle b ^ {n} - (b-1)}$ ${\ displaystyle n \ geq 1}$such that dual Williams primes (or PRP numbers) are of the 1st kind ${\ displaystyle b ^ {n} - (b-1)}$ OEIS link
02 ${\ displaystyle 2 ^ {n} -1}$ 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, 371516667, 4261743801, 43201, 7723288516609, 43201121, 7723288516609, 57232912160 82589933 ... (all Mersenne prime exponents) (Follow A000043 in OEIS )
03 ${\ displaystyle 3 ^ {n} -2}$ 2, 4, 5, 6, 9, 22, 37, 41, 90, 102, 105, 317, 520, 541, 561, 648, 780, 786, 957, 1353, 2224, 2521, 6184, 7989, 8890, 19217, 20746, 31722, 37056, 69581, 195430, 225922, 506233, 761457 ... (Follow A014224 in OEIS )
04th ${\ displaystyle 4 ^ {n} -3}$ 2, 3, 5, 6, 7, 10, 11, 12, 47, 58, 61, 75, 87, 133, 168, 226, 347, 425, 868, 1977, 2815, 3378, 4385, 5286, 7057, 7200, 8230, 8340, 13175, 17226, 18276, 25237, 33211, 58463, 59662, 94555, 120502, 177473, 197017, 351097, 375370 ... (Follow A059266 in OEIS )
05 ${\ displaystyle 5 ^ {n} -4}$ 5, 7, 15, 47, 81, 115, 267, 285, 7641, 19089, 25831, 32115, 59811, 70155 ... (Follow A059613 in OEIS )
06th ${\ displaystyle 6 ^ {n} -5}$ 2, 3, 4, 29, 31, 34, 53, 65, 94, 202, 288, 415, 457, 483, 703, 762, 1285, 1464, 2094, 3384, 9335 ... (Follow A059614 in OEIS )
07th ${\ displaystyle 7 ^ {n} -6}$ 2, 3, 6, 9, 21, 25, 33, 49, 54, 133, 245, 255, 318, 1023, 1486, 3334, 6821, 8555, 11605, 42502, 44409, 90291, 92511, 140303 ... (Follow A191469 in OEIS )
08th ${\ displaystyle 8 ^ {n} -7}$ 13, 661, 773, 833, 4273, 40613 ... (Follow A217380 in OEIS )
09 ${\ displaystyle 9 ^ {n} -8}$ 2, 4, 7, 10, 11, 31, 127, 136, 215, 953, 1139, 1799, 3406, 7633, 13090, 13171, 13511, 32593 ... (Follow A177093 in OEIS )
10 ${\ displaystyle 10 ^ {n} -9}$ 3, 5, 7, 33, 45, 105, 197, 199, 281, 301, 317, 1107, 1657, 3395, 35925, 37597, 64305, 80139, 221631 ... (Follow A095714 in OEIS )

### Dual Williams numbers of the 2nd kind

The smallest , so that it is a prime number, are the following (starting with ): ${\ displaystyle n \ geq 1}$${\ displaystyle b ^ {n} + (b-1) \ in \ mathbb {P}}$${\ displaystyle b = 2}$

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 16, 1, 1, 4, 3, 1, 2, 1, 1, 4, 1, 3, 2, 1, 2, 10, 1, 1, 108, 3, 1, 2, 1, 1, 2, 2, 1, 2, 1, 3, 2, 1, 2, 20, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 4, 2, 2, 7, 8, 3, 1, 2, 1, 24, 2, 1, 1, 12, 4, 3, 8, 1, 1, 4, 3, 1, 194, 3, 1, 2, 1, 2, 2, 1, 8, 2, 1, 1, 4, 2, 2, 54, 1, 1, 4, 1, 1 ... (episode A076845 in OEIS )

It is assumed that there are an infinite number of dual Williams primes of the 2nd kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest dual Williams primes (or -PRP numbers) 2. Type the base with take may (if also solution would be this is in brackets because is actually not allowed, but the completeness led with becomes): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 10}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle b ^ {n} + (b-1)}$ ${\ displaystyle n \ geq 1}$, so that dual Williams primes (or PRP numbers) are of the 2nd kind ${\ displaystyle b ^ {n} + (b-1)}$ OEIS link
02 ${\ displaystyle 2 ^ {n} +1}$ 1, 2, 4, 8, 16 ... (all Fermat prime number exponents)
03 ${\ displaystyle 3 ^ {n} +2}$ (0), 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494, 1131, 1220, 1503, 1858, 4346, 6958, 7203, 10988, 22316, 33508, 43791, 45535, 61840, 95504, 101404, 106143, 107450, 136244, 178428, 361608, 504206 ... (Follow A051783 in OEIS )
04th ${\ displaystyle 4 ^ {n} +3}$ 1, 2, 3, 6, 8, 9, 14, 15, 42, 114, 195, 392, 555, 852, 1004, 1185, 2001, 2030, 2031, 2276, 8610, 8967, 10362, 11366, 15927, 16514, 17877, 19122, 19898, 27728, 29156, 61275, 102981, 117663, 181560, 239922, 342789 ... (Follow A089437 in OEIS )
05 ${\ displaystyle 5 ^ {n} +4}$ (0), 2, 6, 10, 102, 494, 794, 1326, 5242, 5446, 24602, 87606 ... (Follow A124621 in OEIS )
06th ${\ displaystyle 6 ^ {n} +5}$ 1, 2, 4, 7, 10, 14, 18, 32, 55, 102, 177, 190, 247, 276, 372, 1524, 1545, 2502, 4966, 5294, 13030, 13785, 14329, 27333, 44224, 93812 ... (Follow A145106 in OEIS )
07th ${\ displaystyle 7 ^ {n} +6}$ (0), 1, 3, 16, 36, 244, 315, 2577, 9500, 17596, 25551, 32193, 32835, 36504, 75136 ... (Follow A217130 in OEIS )
08th ${\ displaystyle 8 ^ {n} +7}$ 2, 6, 10, 26, 42, 58, 68, 196, 266, 602, 1170, 1288, 1290, 2990, 4110, 6292, 7446, 36928, 57490, 65478, 78570, 188832 ... (Follow A217381 in OEIS )
09 ${\ displaystyle 9 ^ {n} +8}$ 1, 2, 4, 7, 10, 19, 22, 44, 62, 76, 122, 2191, 3134, 9244, 40999, 48230 ... (Follow A217385 in OEIS )
10 ${\ displaystyle 10 ^ {n} +9}$ 1, 2, 3, 4, 9, 18, 22, 45, 49, 56, 69, 146, 202, 272, 2730, 2841, 4562, 31810, 43186, 48109, 92691 ... (Follow A088275 in OEIS )

### Dual Williams numbers of the 3rd kind

The smallest , so that it is a prime number, are the following (starting with ): ${\ displaystyle n \ geq 1}$${\ displaystyle b ^ {n} - (b + 1) \ in \ mathbb {P}}$${\ displaystyle b = 2}$

3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 2, 3, 5, 2, 2, 2, 3, 3, 2, 4, 2, 5, 2, 5, 2, 2, 7, 5, 3, 2, 9, 3, 2, 2, 3, 2, 31, 4, 2, 2, 2, 3, 2, 4, 2, 108, 4, 2, 2, 2, 2, 3, 3, 2, 2, 18, 7, 3, 2, 2, 2, 4, 2, 3, 2, 5, 32, 108, 5, 3, 2, 11, 4, 15, 3, 4, 19, 2, 6, 2, 2, 11, 107, 2, 42, 4, 39, 2, 2, 6, 2, 3 ... (sequence A178250 in OEIS )

It is assumed that there are an infinite number of dual Williams primes of the 3rd kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest dual Williams primes (or -PRP numbers) 3. Type the base with take can: ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 10}$

${\ displaystyle b}$ ${\ displaystyle b ^ {n} - (b + 1)}$ ${\ displaystyle n \ geq 1}$, so that dual Williams primes (or PRP numbers) are kind 3 ${\ displaystyle b ^ {n} - (b + 1)}$ OEIS link
02 ${\ displaystyle 2 ^ {n} -3}$ 3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452, 36552, 42689, 44629, 50474, 66422, 69337, 116926, 119324, 123297, 189110, 241004, 247165, 284133, 354946, 394034, 702194, 750740, 840797, 1126380, 1215889, 1347744 ... (Follow A050414 in OEIS )
03 ${\ displaystyle 3 ^ {n} -4}$ 2, 3, 5, 21, 31, 37, 41, 53, 73, 101, 175, 203, 225, 455, 557, 651, 1333, 4823, 20367, 32555, 52057, 79371, 267267, 312155 ... (Follow A058959 in OEIS )
04th ${\ displaystyle 4 ^ {n} -5}$ 2, 3, 4, 5, 6, 9, 10, 13, 16, 18, 28, 33, 59, 65, 75, 83, 103, 113, 275, 353, 405, 568, 614, 909, 1184, 1200, 1564, 2266, 2556, 4246, 8014, 8193, 8696, 9291 ... (Follow A217348 in OEIS )
05 ${\ displaystyle 5 ^ {n} -6}$ 2, 4, 5, 6, 10, 53, 76, 82, 88, 242, 247, 473, 586, 966, 1015, 1297, 1825, 2413, 2599, 2833, 5850, 5965, 6052, 27199, 49704, 79000 ... (Follow A165701 in OEIS )
06th ${\ displaystyle 6 ^ {n} -7}$ 2, 4, 6, 8, 9, 10, 15, 20, 46, 49, 61, 98, 110, 144, 266, 344, 978, 1692, 1880, 1924, 3142, 3220, 4209, 5708, 7064 ... (Follow A217352 in OEIS )
07th ${\ displaystyle 7 ^ {n} -8}$ 2, 4, 8, 10, 50, 106, 182, 293, 964, 1108, 1654, 1756, 4601, 8870, 15100, 17446, 22742, 34570, 50150, 95276 ... (Follow A217131 in OEIS )
08th ${\ displaystyle 8 ^ {n} -9}$ 3, 7, 11, 47, 81, 95, 107, 179, 233, 243, 947, 2817, 2859, 3233, 7563, 11307 ... (Follow A217383 in OEIS )
09 ${\ displaystyle 9 ^ {n} -10}$ 2, 3, 4, 9, 11, 18, 19, 27, 28, 46, 50, 53, 80, 155, 203, 280, 451, 4963 ... (Follow A217493 in OEIS )
10 ${\ displaystyle 10 ^ {n} -11}$ 2, 5, 8, 12, 15, 18, 20, 30, 80, 143, 152, 164, 176, 239, 291, 324, 504, 594, 983, 2894, 22226, 35371, 58437, 67863, 180979 ... (Follow A092767 in OEIS )

### Dual Williams numbers of the 4th kind

The following applies: There is no such thing as a dual Williams prime number of the 4th type with a base . ${\ displaystyle b \ equiv 1 {\ pmod {3}}}$

Proof:
If is, also applies . Furthermore is . Thus one obtains . So in this case what was to be shown is always divisible by and thus never a prime number.${\ displaystyle b \ equiv 1 {\ pmod {3}}}$${\ displaystyle b ^ {n} \ equiv 1 ^ {n} = 1 {\ pmod {3}}}$${\ displaystyle b + 1 \ equiv 1 + 1 = 2 {\ pmod {3}}}$${\ displaystyle b ^ {n} + (b + 1) \ equiv 1 + 2 = 3 \ equiv 0 {\ pmod {3}}}$${\ displaystyle b ^ {n} + (b + 1)}$${\ displaystyle 3}$${\ displaystyle \ Box}$

It is assumed that there are an infinite number of dual Williams primes of the 4th kind for the base . ${\ displaystyle b}$

Below is a table which is the smallest dual Williams primes (or -PRP numbers) 4. Type the base with take may (if also solution would be this is in brackets because is actually not allowed, but the completeness led with becomes): ${\ displaystyle b}$${\ displaystyle 2 \ leq b \ leq 10}$${\ displaystyle n = 0}$${\ displaystyle n = 0}$

${\ displaystyle b}$ ${\ displaystyle b ^ {n} + (b + 1)}$ ${\ displaystyle n \ geq 1}$so that dual Williams primes (or PRP numbers) are of the 4th kind ${\ displaystyle b ^ {n} + (b + 1)}$ OEIS link
02 ${\ displaystyle 2 ^ {n} +3}$ 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947 ... (Follow A057732 in OEIS )
03 ${\ displaystyle 3 ^ {n} +4}$ (0), 1, 2, 3, 6, 9, 10, 22, 30, 42, 57, 87, 174, 195, 198, 562, 994, 2421, 2487, 4629, 5838, 13698, 14730, 16966, 25851, 98634, 117222, 192819 ... (Follow A058958 in OEIS )
04th ${\ displaystyle 4 ^ {n} +5}$ there are no prime numbers because of this form ${\ displaystyle b = 4 \ equiv 1 {\ pmod {3}}}$
05 ${\ displaystyle 5 ^ {n} +6}$ (0), 1, 2, 3, 4, 13, 88, 177, 184, 297, 304, 310, 562, 892, 1300, 4047, 5557, 9028, 15597, 28527, 56890, 77485, 79378 ... (Follow A089142 in OEIS )
06th ${\ displaystyle 6 ^ {n} +7}$ 1, 2, 3, 4, 6, 21, 24, 27, 30, 54, 70, 126, 369, 435, 612, 787, 1275, 2155, 2436, 5734, 6016 ... (Follow A217351 in OEIS )
07th ${\ displaystyle 7 ^ {n} +8}$ there are no prime numbers because of this form ${\ displaystyle b = 7 \ equiv 1 {\ pmod {3}}}$
08th ${\ displaystyle 8 ^ {n} +9}$ 1, 2, 3, 6, 10, 19, 22, 109, 798, 1498, 1519, 3109, 5491, 13351, 26983 ... (Follow A217382 in OEIS )
09 ${\ displaystyle 9 ^ {n} +10}$ (0), 1, 3, 4, 9, 18, 49, 57, 67, 69, 106, 126, 258, 583, 1221, 1366, 4311 ... (Follow A217492 in OEIS )
10 ${\ displaystyle 10 ^ {n} -11}$ there are no prime numbers because of this form ${\ displaystyle b = 10 \ equiv 1 {\ pmod {3}}}$