Mersenne prime

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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 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

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.