Woodall number

In number theory, a Woodall number (W_n) is any natural number of the form

[math]\displaystyle{ W_n = n \cdot 2^n - 1 }[/math]

for some natural number n. The first few Woodall numbers are:

1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS).

History

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly defined Cullen numbers.

Woodall primes

Unsolved problem in mathematics: Are there infinitely many Woodall primes? (more unsolved problems in mathematics)

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. In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers n · 2^{n + a} + b, where a and b are integers, and in particular, that almost all Woodall numbers are composite. It is an open problem whether there are infinitely many Woodall primes. (As of October 2018), the largest known Woodall prime is 17016602 × 2^17016602 − 1. It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.

Restrictions

Starting with W₄ = 63 and W₅ = 159, every sixth Woodall number is divisible by 3; thus, in order for W_n to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W_2^m may be prime only if 2^m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W₂ = M₃ = 7, and W₅₁₂ = M₅₂₁.

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 [math]\displaystyle{ \left(\frac{2}{p}\right) }[/math] is +1 and

W_{(3p − 1) / 2} if the Jacobi symbol [math]\displaystyle{ \left(\frac{2}{p}\right) }[/math] is −1.

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

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), the largest known generalized Woodall prime with base greater than 2 is 2740879 × 32^2740879 − 1.

b	Numbers n such that n × bn − 1 is prime	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.

Further reading

• 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", Mathematics of Computation 64 (212): 1733–1741, doi:10.2307/2153382, http://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf .
• Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, http://primes.utm.edu/top20/page.php?id=7, 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". http://mathworld.wolfram.com/WoodallNumber.html. 
• Steven Harvey, List of Generalized Woodall primes.
• Paul Leyland, Generalized Cullen and Woodall Numbers