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 instance the Solovay–Strassen primality test on a given number formula_20 chooses a number formula_34 randomly from formula_42 to formula_210 and uses modular exponentiation to check whether formula_211 is divisible by formula_20. If so, it answers yes and otherwise it answers no. If formula_20 really is prime, it will always answer yes, but if formula_20 is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. If this test is repeated formula_1 times on the same number, the probability that a composite number could pass the test every time is at most formula_216. Because