# Woodall number

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

Wn = n × 2n − 1

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

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

Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,[citation needed] inspired by James Cullen's earlier study of the similarly-defined Cullen numbers. Woodall numbers curiously arise in Goodstein's theorem.[citation needed]

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 OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence A050918 in OEIS).

In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.[citation needed] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n · 2n+a + b where a and b are integers,[citation needed] and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes.[citation needed] As of December 2007, the largest known Woodall prime is 3752948 × 23752948 − 1.[1] It has 1,129,757 digits and was found by Matthew J. Thompson in 2007 in the distributed computing project PrimeGrid.

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 $\left(\frac{2}{p}\right)$ is +1 and
W(3p − 1) / 2 if the Jacobi symbol $\left(\frac{2}{p}\right)$ is −1.[citation needed]

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.