# Mersenne prime

From Number

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.

View other properties of prime numbers | View other properties of natural numbers

## Contents

## Definition

A **Mersenne prime** is a Mersenne number that is also a prime number. In other words, it is a number of the form that is prime, where is a natural number.

It turns out that if is prime, then itself is also prime, though the converse is not true (the smallest counterexample is , because ).

## Facts

### Facts in number theory

- Mersenne number is prime implies number is prime
- Mersenne number for prime or Poulet implies prime or Poulet

### Facts in other branches of mathematics

## Occurrence

### Initial examples

The Mersenne numbers are prime for , with the corresponding primes being . Them smallest prime for which the Mersenne number is *not* prime is : .

### Infinitude conjecture

`Further information: Infinitude conjecture for Mersenne primes`

It is conjectured that there are infinitely many Mersenne primes.