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 a 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 39 Mersenne primes known (view a list of them), and more are being discovered from time to time. A lot of is currently being done by the Great Internet Mersenne Prime Search. The larger Mersenne primes are the largest numbers which we know to be prime.

The n^th 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.