Definition:Mersenne Number

Definition

A Mersenne number is a natural number of the form $2^p - 1$, where $p$ is prime.

The number $2^p - 1$ is, in this context, often denoted $M_p$.

They are named for Marin Mersenne, who wrote a book on the subject and aroused considerable interest in their study.

Factors of Mersenne Numbers

From Factors of Mersenne Numbers, we see that all divisors of $2^p - 1$ are of the form $2 k p + 1$.

Thus any factors of Mersenne numbers can conveniently be referred to by the value of $k$.

Also see

This sequence is A000225 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

When a Mersenne number is prime, it is called a Mersenne Prime.

• Also see the Lucas-Lehmer Test.