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In mathematics, a **Mersenne prime** is a prime number that is one less than a power of two. That is, it is a prime number of the form Template:Math for some integer Template:Math. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.

The exponents Template:Math which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... OEIS A{{{1}}} and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... OEIS A{{{1}}}.

If Template:Math is a composite number then so is Template:Math. (Template:Math is divisible by both Template:Math and Template:Math.) This definition is therefore equivalent to the definition as a prime number of the form Template:Math for some prime Template:Math.

More generally, numbers of the form Template:Math without the primality requirement may be called **Mersenne numbers**. Sometimes, however, Mersenne numbers are defined to have the additional requirement that Template:Math be prime.
The smallest composite Mersenne number with prime exponent *n* is Template:Nowrap.

Mersenne primes Template:Math are also noteworthy due to their connection to perfect numbers.

Template:As of, 51 are now known. The largest known prime number Template:Nowrap is a Mersenne prime.^{[1]} 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