: Mathematics: Mersenne prime

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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 $M n = 2 n - 1$ for some integer $n$ . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If $n$ is a composite number then so is $2 n - 1$ . 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 $p$ .
The exponents $n$ which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).
Numbers of the form $M n = 2 n - 1$ without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that $n$ be prime.
The smallest composite Mersenne number with prime exponent n is 2¹¹ − 1 = 2047 = 23 × 89.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Mersenne_prime)

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