# 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.
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## 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 ).

## 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.