Woodall number
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In mathematics, a Woodall number is a natural number of the form n · 2n − 1 (written Wn). Woodall numbers were first studied by A. J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly-defined Cullen numbers. The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, ... Template:OEIS. Woodall numbers curiously arise in Goodstein's theorem.
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, ... Template:OEIS; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... Template:OEIS.
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>\left(\frac{2}{p}\right)</math> is +1 and
- W(3p − 1) / 2 if the Jacobi symbol <math>\left(\frac{2}{p}\right)</math> is −1.
It is conjectured that almost all Woodall numbers are composite; a proof has been submitted by Suyama, but it has not been verified yet. Nonetheless, it is also conjectured that there are infinitely many Woodall primes.
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.