Definition:Cullen Prime

Definition

A Cullen prime is a Cullen 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:

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

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

• Definition:Cullen Number

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: pp. 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. https://mathworld.wolfram.com/CullenNumber.html