|Discovered by||Marin Mersenne|
|Number of||Conjectured infinite|
|Description||A prime which is one less than a power of two|
|First Few||3, 7, 31, 127, 8,191 , 131,071|
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 2756,839 − 1 was proved prime, the largest known prime number has always been a Mersenne Prime. In 2014, 257,885,161 − 1 was found to be prime, which contains 17,425,170 digits. As of 2016, the largest known prime is 274,207,281 -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, 2p - 1, the composite number (2p-1)x2p-1 is perfect.