Mersenne Primes are numbers that can be expressed in the form 2^{p} − 1, where p is a prime number. Not all numbers of the form 2^{p} − 1 are prime, but those which are prime are known as Mersenne primes, named after French mathematician, Marin Mersenne. Numbers of the form 2^{n} − 1 where n is composite cannot be prime.

Since 1992, when 2^{756,839} − 1 was proved prime, the largest known prime number has always been a Mersenne Prime. In 2014, 2^{57,885,161} − 1 was found to be prime, which contains 17,425,170 digits. As of 2016, the largest known prime is 2^{74,207,281 }-1 which has 22,338,618 digits.

Mersenne Primes are closely linked with Perfect numbers (numbers which are the sum of their proper divisors). For any Mersenne Prime, 2^{p} - 1, the composite number (2^{p}-1)x2^{p-1} is perfect.