# Definition:Woodall Prime

## Definition

A **Woodall prime** is a Woodall number:

- $n \times 2^n - 1$

which is also prime.

### Sequence

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

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

This sequence is A002234 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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

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

This sequence is A050918 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also known as

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**.

## Also see

## Source of Name

This entry was named for Herbert J. Woodall.

## Sources

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