A Mersenne number (written M_{p}) is a number of the
form 2^{ p} − 1.
Mersenne primes are prime Mersenne numbers.
If 2^{ p} − 1 is prime, then `p` itself must also be prime;
when `p` is composite, it can be shown that 2^{ p} − 1
is always composite.

In 1644, Father Marin
Mersenne, a natural philosopher, theologian, mathematician, and
musical theorist, claimed in the preface to
Cogitata Physico-Mathematica
that the only values of `p` no greater than 257 for which 2^{ p} − 1
is prime are 1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. There are
a few errors on this list; M_{61} is prime while M_{67}
is composite (though this might have been a typo); M_{257} is
composite while M_{89} and M_{107} are prime.

Nonetheless, this was an amazing accomplishment. There were no computers to perform calculations in those days. As well, this list has provided a stimulus to mathematicians to invent better methods of factoring (in order to check whether a given Mersenne number is prime or not).

Currently, there are 48 Mersenne primes known (view a list of them), and more are being discovered from time to time, many of which are being discovered through the Great Internet Mersenne Prime Search. The larger Mersenne primes are the largest numbers which we know to be prime.

The `n`^{th} even perfect number is
given by M_{p} × (2^{ n − 1}), where M_{p} is
a Mersenne prime. Since all even perfect numbers are given by this
formula, there is a one-to-one correspondence between even perfect
numbers and Mersenne primes.