Mersenne prime 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 for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. 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, ... . If is a composite number then so is . ( is divisible by both and .)
Prime number is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime formula_20 is expressible as the sum of two squares, formula_187, and therefore factorizable as formula_188, exactly when formula_20 is 1 mod 4. Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_190 (for integers formula_34 and formula_35) the number formula_193 has two factorizations formula_194, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion