Woodall number
In number theory, a Woodall number (Wn) is any natural number of the form
for some natural number n. The first few Woodall numbers are:
History
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
Are there infinitely many Woodall primes?
Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn 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
Restrictions
Starting with W4 = 63 and W5 = 159, every sixth Woodall number is
Divisibility properties
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
A generalized Woodall number base b is defined to be a number of the form n × bn − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime.
The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... are[6]
- 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)
As of November 2021[update], the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1.[7]
b | Numbers n such that n × bn − 1 is prime[6] | OEIS sequence
|
---|---|---|
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, 524427, ... | A059671 |
11 | 2, 8, 252, 1184, 1308, 1182072, ... | A299374 |
12 | 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, 549721, 866981, 1405486, ... | A299375 |
13 | 2, 6, 563528, ... | A299376 |
14 | 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, 1167708, ... | A299377 |
15 | 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, 1527090, ... | 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, 663703, ... | A299383 |
See also
- Mersenne prime - Prime numbers of the form 2n − 1.
References
- Cunningham, A. J. C; Woodall, H. J.(1917), "Factorisation of and ", Messenger of Mathematics, 47: 1–38.
- Zbl 1033.11006.
- ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archivedfrom the original on February 28, 2020. Retrieved October 1, 2020.
- ^ "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018
- ^ PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018
- ^ a b List of generalized Woodall primes base 3 to 10000
- ^ "The Top Twenty: Generalized Woodall". primes.utm.edu. Retrieved 20 November 2021.
Further reading
- ISBN 0-387-20860-7.
- Keller, Wilfrid (1995), "New Cullen Primes" (PDF), JSTOR 2153382.
- Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007.
External links
- 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