Woodall prime
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:
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 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 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, … (sequence in the OEIS).
In 1976 Christopher Hooley showed that almost all Cullen numbers are composite. 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, and in particular also for Woodall numbers. Nonetheless, it is conjectured that there are infinitely many Woodall primes. As of February 2016[update], the largest known Woodall prime is 3752948 × 23752948 − 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
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.
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