====== Mersenne primes ======

//Mersenne Primes// are a specific case of [[content:mathematics:prime_numbers]]first described by the French mathematician Martin Mersenne in the early 17th century.

They take the form of //M<sub>n</sub>// = 2//<sup>n</sup>// − 1

i.e. a prime number that is one less than a power of two. For example, 31, which is  2<sup>5</sup> − 1.

It's not currently known if there are an infinite number of //Mersenne Primes. //To date, only 51 have been discovered. The search is significantly driven by a distributed computing project known as the [[https://www.mersenne.org/| Great Internet Mersenne Prime Search]].

It's also not known whether infinitely many //Mersenne numbers// with prime exponents are //composite//, i.e. they can be formed by multiplying two smaller positive integers.

There are no theorems for predicting the next //Mersenne Primes//, though there are conjectures about their distribution. See : [[https://primes.utm.edu/mersenne/heuristic.html|PrimePages]] , University of Tennessee at Martin.