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 $2^p - 1$.

From Primes of form Power Less One, it is clear that in order for $2^p - 1$ to be prime, then $p$ must also be prime.

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

It is not known whether there is an infinite number of Mersenne primes.

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.

G.I.M.P.S.
"G.I.M.P.S." (Great Internet Mersenne Prime Search), or just "GIMPS", has become a gathering place for Number Theorists interested in the discovery of the Mersenne primes.

In August and September of $2008$ alone, both the $45$th and $46$th Mersenne primes were discovered:

$M_{43,112,609}$ (a $12,978,189$ digit number)

and

$M_{37,156,667}$ (an $11,185,272$ digit number)

becoming the first Mersenne primes of $10$ million digits to be found.

You can follow the work of G.I.M.P.S. at www.mersenne.org.