Woodall number

In mathematics, a Woodall number (also called Riesel number) is a natural number of the form n · 2ⁿ - 1 (written W_n). Woodall numbers were first studied by A. J. C. Cunningham and H. J. Woodall in 1917, inspired by J. Cullen's earlier study of the similarly-defined Cullen numbers. The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, ... (Sloane's A003261).

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, ... (Sloane's A002234); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (Sloane's A050918).

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 (2|p) is +1 and W_{(3p - 1) / 2} if the Jacobi symbol (2|p) is -1. It is conjectured that almost all Woodall numbers are composite; a proof has been submitted by Suyama, but not verified yet.

A generalized Woodall number 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.

External links

• The Prime Glossary: Woodall number
• MathWorld: Woodall number