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 examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants formula_51 and formula_52 such that are prime for any natural number formula_1 in the first formula, and any number of exponents in the second formula. Here formula_55 represents the floor function, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of formula_56 or formula_52. Many conjectures revolving about primes have been posed. Often having an elementary