Mersenne prime

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

Definition

A Mersenne prime is a Mersenne number that is also a prime number. In other words, it is a number of the form $M_{n}=2^{n}-1$ that is prime, where $n$ is a natural number.

It turns out that if $M_{n}$ is prime, then $n$ itself is also prime, though the converse is not true (the smallest counterexample is $n=11$, because $M_{11}=2047=23\cdot 89$).

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

• Order is product of Mersenne prime and one more implies normal Sylow subgroup (fact about groups)

Occurrence

Initial examples

The Mersenne numbers $M_{p}$ are prime for $p=2,3,5,7$, with the corresponding primes $M_{p}$ being $3,7,31,127$. Them smallest prime $p$ for which the Mersenne number $M_{p}$ is not prime is $11$: $M_{11}=2047=23\cdot 89$.

Infinitude conjecture

Further information: Infinitude conjecture for Mersenne primes

It is conjectured that there are infinitely many Mersenne primes.