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 = 2^{1}(2^{2}-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_{i}^{2} - 2 (mod M_{q}), starting from S_{0} = 4 . if S_{q-2} = 0 (mod M_{q}) , then M_{q} is prime.
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