Definition:Woodall Prime

Definition

A Woodall prime is a Woodall number:

$n \times 2^n - 1$

which is also prime.

The sequence $\sequence n$ for which $n \times 2^n - 1$ is a prime number begins:

$2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, \ldots$

The first few of these correspond with the sequence $\sequence n$ of the actual Woodall primes themselves, which begins:

$7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, \ldots$

Some sources refer to primes of the form $n \times 2^n - 1$ as Cullen primes, along with those of the form $n \times 2^n + 1$.

However, it is now conventional to reserve the term Cullen primes, named for James Cullen, to those of the form $n \times 2^n + 1$.

The latter are also known as Cunningham primes, for Allan Joseph Champneys Cunningham, so as to ensure their unambiguous distinction from Woodall primes.

This entry was named for Herbert J. Woodall.

Weisstein, Eric W. "Woodall Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WoodallNumber.html