Mersenne prime

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 2^''n'' − 1}} for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form 2^''p'' − 1}} for some prime .

The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... .

Numbers of the form 2^''n'' − 1}} without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that should be prime. The smallest composite Mersenne number with prime exponent ''n'' is 2047 23 × 89}}.

Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

, 52 Mersenne primes are known. The largest known prime number, , is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once. Provided by Wikipedia