Mersenne Primes

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

Since 1992, when 2756839 − 1 was proved prime, the largest known prime number has always been a Mersenne Prime. In 2014,  257885161 − 1 was found to be prime, which contains 17425170 digits. As of 2016, the largest known prime is 274207281-1 which has 22338618 digits.

Mersenne Primes are closely linked with Perfect numbers (numbers which are the sum of their proper divisors). For any Mersenne Prime, 2p - 1, the composite number (2p-1)x2p-1 is perfect.

The First Few Mersenne Prime Numbers
3

7

31

127

8191

131071

524287