In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two:

Some definitions of Mersenne numbers require that the exponent p be prime, since the associated number must be composite if p is composite.

A Mersenne prime is a Mersenne number that is prime. It is known that if 2^p − 1 is prime then p is prime, so it makes no difference which Mersenne number definition is used. As of October 2009^[ref], 47 Mersenne primes are known. The largest known prime number (2^43,112,609 – 1) is a Mersenne prime. Since 1997, all newly-found Mersenne primes have been discovered by the "Great Internet Mersenne Prime Search" (GIMPS), a distributed computing project on the Internet.

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite. The Lenstra–Pomerance–Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3 (mod 4).