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 the notion that the "density" of primes around should be . This function is related to the logarithm by the asymptotic expansion So, the prime number theorem can also be written as . In fact, in another paper in 1899 de la Vallée Poussin proved that for some positive constant , where is the big notation. This has been improved to Because of the connection between the Riemann zeta function and , the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than