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 function that gives the number of primes less than or equal to , for any real number . For example, because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that is a good approximation to , in the sense that the limit of the "quotient" of the two functions and as increases without bound is 1: known as the asymptotic law of distribution of prime numbers. Using asymptotic notation this result can be restated as This notation (and the theorem) does "not" say anything about the