Woodall number Totally Explained

Everything about Woodall Prime totally explained

In mathematics, a Woodall number is a natural number of the form n · 2ⁿ − 1 (written W_n). 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. The first few Woodall numbers are 1, 7, 23, 63, 159, 383, 895, ... .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 W_n are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... ; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... .
Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides ť W_{(p + 1) / 2} if the Jacobi symbol

left(frac
ight)

is −1.

It is conjectured that almost all Woodall numbers are composite; a proof has been submitted by H. Suyama, but it hasn't been verified yet. Nonetheless, it's also conjectured that there are infinitely many Woodall primes. As of December 2007, the largest known Woodall prime is 3752948 · 2^3752948 − 1. It has 1129757 digits and was found by Matthew J Thompson in 2007 in the distributed computing project PrimeGrid.
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's then called a generalized Woodall prime.

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