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 also a sufficient condition for formula_20 to be prime. Wilson's theorem says that an integer formula_145 is prime if and only if the factorial formula_146 is congruent to formula_147 mod formula_20. For a composite this cannot hold, since one of its factors divides both and formula_149, and so formula_150 is impossible. The formula_20-adic order formula_152 of an integer formula_1 is the number of copies of formula_20 in the prime factorization of formula_1. The same concept can be extended from integers to rational numbers by defining the formula_20-adic order of a fraction formula_157 to be formula_158. The formula_20-adic absolute value