Definition:Mersenne Number

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

The number $2^p - 1$, in this context, can be denoted $M_p$.

Also defined as
A Mersenne number can also be seen defined as a natural number of the form $2^n - 1$, where $n \in \Z_{\ge 0}$ or $n \in \Z_{> 0}$, but that leads to the singularly boring sequence:
 * $\left({0,}\right) 1, 3, 7, 15, 31, 63, 127, 255, 511, 1 \, 023, 2 \, 047, 4 \, 095, 8 \, 191, 16 \, 383, 32 \, 767, \ldots$

Also see

 * Factors of Mersenne Numbers, where it is shown that 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 $k$.


 * Definition:Mersenne Prime
 * Lucas-Lehmer Test