Woodall number
In number theory, a Woodall number (W_{n}) is any natural number of the form
for some natural number n. The first few Woodall numbers are:
History[edit]
Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,^{[1]} inspired by James Cullen's earlier study of the similarly-defined Cullen numbers.
Woodall primes[edit]
Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers W_{n} are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, … (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence A050918 in the OEIS).
In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.^{[2]} Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2^{n+a} + b where a and b are integers, and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.^{[citation needed]} As of March 2018^{[update]}, the largest known Woodall prime is 17016602 × 2^{17016602} − 1.^{[3]} It has 5,122,515 digits and was found by Diego Bertolotti in 2018 in the distributed computing project PrimeGrid.^{[4]}
Divisibility properties[edit]
Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides
- W_{(p + 1) / 2} if the Jacobi symbol is +1 and
- W_{(3p − 1) / 2} if the Jacobi symbol is −1.^{[citation needed]}
Generalization[edit]
A generalized Woodall number base b is defined to be a number of the form n × b^{n} − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime.
Least n such that n × b^{n} - 1 is prime are^{[5]}
- 3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ... (sequence A240235 in the OEIS)
b | numbers n such that n × b^{n} - 1 is prime (these n are checked up to 350000) | OEIS sequence |
1 | 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, ... (all primes plus 1) | A008864 |
2 | 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, ... | A002234 |
3 | 1, 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ... | A006553 |
4 | 1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... | A086661 |
5 | 8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... | A059676 |
6 | 1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... | A059675 |
7 | 2, 18, 68, 84, 3812, 14838, 51582, ... | A242200 |
8 | 1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... | A242201 |
9 | 10, 58, 264, 1568, 4198, 24500, ... | A242202 |
10 | 2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ... | A059671 |
11 | 2, 8, 252, 1184, 1308, ... | A299374 |
12 | 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ... | A299375 |
13 | 2, 6, 563528, ... | A299376 |
14 | 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ... | A299377 |
15 | 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ... | A299378 |
16 | 167, 189, 639, ... | A299379 |
17 | 2, 18, 20, 38, 68, 3122, 3488, 39500, ... | A299380 |
18 | 1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... | A299381 |
19 | 12, 410, 33890, 91850, 146478, 189620, 280524, ... | A299382 |
20 | 1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ... | A299383 |
21 | 2, 18, 200, 282, 294, 1174, 2492, 4348, ... | |
22 | 2, 5, 140, 158, 263, 795, 992, 341351, ... | |
23 | 29028, ... | |
24 | 1, 2, 5, 12, 124, 1483, 22075, 29673, 64593, ... | |
25 | 2, 68, 104, 450, ... | |
26 | 3, 8, 79, 132, 243, 373, 720, 1818, 11904, 134778, ... | |
27 | 10, 18, 20, 2420, 6638, 11368, 14040, 103444, ... | |
28 | 2, 5, 6, 12, 20, 47, 71, 624, 1149, 2399, 8048, 30650, 39161, ... | |
29 | 26850, 237438, 272970, ... | |
30 | 1, 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, 201038, ... |
As of September 2017^{[update]}, the largest known generalized Woodall prime is 1993191×4^{1993191} − 1.
See also[edit]
- Mersenne prime - Prime numbers of the form 2^{n} − 1.
References[edit]
- ^ Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of and ", Messenger of Mathematics, 47: 1–38.
- ^ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006.
- ^ "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018
- ^ PrimeGrid, Anouncement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018
- ^ List of generalized Woodall primes base 3 to 10000
Further reading[edit]
- Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7.
- Keller, Wilfrid (1995), "New Cullen Primes" (PDF), Mathematics of Computation, 64 (212): 1733–1741, doi:10.2307/2153382.
- Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007.
External links[edit]
- Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The Top Twenty: Generalized Woodall, at The Prime Pages.
- Weisstein, Eric W. "Woodall number". MathWorld.
- Steven Harvey, List of Generalized Woodall primes.
- Paul Leyland, Generalized Cullen and Woodall Numbers