A Mersenne prime is a prime that is in the form of , where is an integer. It is named after Marin Mersenne.
These are some of the largest primes known to man due to one main factor: There is an integer bit value set to that, so that the largest number with a certain amount of bits is a form of .
For example: The amount of numbers on a 32 bit computer is . Then, divide by 2, as there are positive, and negative values. Then subtract one, as zero is one of them, so the largest number on a 32 bit computer is 2,147,483,647. (Not necessarily the largest number displayed, to achieve a higher number, a computer could use a base system other than 2.)
As of January 2018, the largest known prime is , a Mersenne prime which contains 23,249,425 digits.
Connection with Even Perfect Numbers
All even perfect numbers are of the form where is a Mersenne prime, which was proven by Euler in the 18th century.