Mersenne prime

A Mersenne prime is a prime that is in the form of $2^n-1$ , where $n$ 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 $2^n-1$ .

For example: The amount of numbers on a 32 bit computer is $2^{32}$ . 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 $2^{77,232,917}-1$ , a Mersenne prime which contains 23,249,425 digits.

Connection with Even Perfect Numbers

All even perfect numbers are of the form $\frac{p(p+1)}{2}$ where $p = 2^k-1$ is a Mersenne prime, which was proven by Euler in the 18th century.

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Mersenne prime

Connection with Even Perfect Numbers