Woodall number

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 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] 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.[3] 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.[4] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.[5]

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.

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 ${\displaystyle \left({\frac {2}{p}}\right)}$ is +1 and

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

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, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 32^2740879 − 1.[7]

• Mersenne prime - Prime numbers of the form 2ⁿ − 1.

• 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, JSTOR 2153382.
• Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007.

• 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