Mersenne Primes

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; M₆₁ is prime while M₆₇ is composite (though this might have been a typo); M₂₅₇ is composite while M₈₉ and M₁₀₇ 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.