---
tag: 06GPJF9V7M
kind: definition
name: Mersenne prime
msc: [11A41]
defines: 0F3Z22M2GT
uses: [0GJBJ19NXR, 0SN0MAEKYX]
sources:
  - url: https://en.wikipedia.org/wiki/Mersenne_prime
    locator: About Mersenne primes
---

## Definition

A Mersenne prime is a [[prime number|0SN0MAEKYX]] of the form $2^n - 1$ for a natural number $n$.
That is, a Mersenne prime is a [[Mersenne number|0GJBJ19NXR]] 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|07MMWAXMYD]]. The converse fails: $2^{11} - 1 = 2047 = 23 \cdot 89$.
The [[Lucas–Lehmer test|061BTYA0RN]] 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|0SB7Q8SYSF]] come exactly from these primes, by the [[Euclid–Euler theorem|0WETJCKNRG]].