Definition:Mersenne Prime

Definition
A Mersenne prime is a Mersenne number which happens to be prime.

That is, it is a prime number of the form $2^p - 1$.

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

Testing Primality of a Mersenne number
The Lucas-Lehmer Test is a way of determining the primality of a given $M_p$ without laboriously testing each possible prime divisor.

Also see

 * Definition:Mersenne Number
 * Theorem of Even Perfect Numbers
 * Primes of form Power Less One: for $2^p - 1$ to be prime, then $p$ must also be prime.
 * Do Mersenne Primes Go On For Ever?