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. The smallest composite Mersenne number with prime exponent n is 2 11 − 1 = 2047 = 23 × 89. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. Numbers of the form M n = 2 n − 1 without the primality requirement may be called Mersenne numbers. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form M p = 2 p − 1 for some prime p. If n is a composite number then so is 2 n − 1. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. That is, it is a prime number of the form M n = 2 n − 1 for some integer n. In mathematics, a Mersenne prime is a prime number that is one less than a power of two. Mersenne primes (of form 2^ p − 1 where p is a prime).
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