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 theorem Dusart proved: The proof by de la Vallée Poussin implies the following. For every , there is an such that for all , As a consequence of the prime number theorem, one gets an asymptotic expression for the th prime number, denoted by : A better approximation is Again considering the th prime number , this gives an estimate of ; the first 5 digits match and relative error is about 0.00005%. Rosser's theorem states that This can be improved by the following pair of bounds: The table compares exact values of to the two approximations and . The last