Mersenne prime

Definition

A Mersenne prime is a prime number of the form $2^n - 1$ for a natural number $n$. That is, a Mersenne prime is a Mersenne number that is prime.

Remarks

The first Mersenne primes are $3$, $7$, $31$ and $127$, for $n = 2, 3, 5, 7$. If $2^n - 1$ is prime, then $n$ is prime. The converse fails: $2^{11} - 1 = 2047 = 23 \cdot 89$. The Lucas–Lehmer test decides whether $2^p - 1$ is prime. It is not known whether there are infinitely many Mersenne primes.

Explanations

L1 human

A Mersenne prime is a prime that is one less than a power of $2$, such as $31 = 2^5 - 1$. The even perfect numbers come exactly from these primes, by the Euclid–Euler theorem.