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 $M_{n} = 2^{n} - 1$ 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 $M_{11} = 2047 = 23 \cdot 89$ ).

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.