# Definition:Cullen Prime

## Contents

## Definition

A **Cullen prime** is a Cullen number:

- $n \times 2^n + 1$

which is also prime.

### Sequence

The sequence $\left\langle{n}\right\rangle$ for which $n \times 2^n + 1$ is a prime number begins:

- $1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, \ldots$

This sequence is A005849 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Also defined as

Some sources refer to primes of the form $n \times 2^n - 1$ as also being **Cullen primes**.

However, it is now conventional to refer to numbers of the form $n \times 2^n - 1$ as **Woodall primes**, for Herbert J. Woodall.

## Also known as

Some sources refer to **Cullen primes** as **Cunningham primes**, for Allan Joseph Champneys Cunningham, so as to ensure their distinction from **Woodall primes**.

## Also see

## Source of Name

This entry was named for James Cullen.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $141$ - Oct. 1995: Wilfrid Keller:
*New Cullen Primes*(*Math. Comp.***Vol. 64**,*no. 212*: 1733 – 1741) www.jstor.org/stable/2153382 - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $141$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $4713$

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