Mersenne prime

Mersenne primes are a category of numbers studied by the monk Marin Mersenne (1588-1648) and by Pierre de Fermat (1601-1665), and defined by: M_q = 2^q - 1 where q is a prime number.

Every Mersenne prime can be used to construct an even perfect number 2^q-1 M_q , equal to the sum of its divisors (example: 6 = 2¹(2²-1) = 1+2+3).

It is possible to prove that a Mersenne prime is indeed prime by a test invented by Edouard Lucas (1842-1891) and rigorously proved by Derrick Lehmer (1905-1991): the LLT test (Lucas-Lehmer-Test).

This test consists in calculating the elements of the series: S_i+1 = S_i² - 2 (mod M_q), starting from S₀ = 4 . if S_q-2 = 0 (mod M_q) , then M_q is prime.