A Mersenne number is a number of the from 2n − 1, where n is a positive
integer. If Mersenne number is prime, then it is called a Mersenne prime. It is
well known that if 2n − 1 is prime, then n is also prime.
Mersenne numbers have played an important role in number theory since
1644. In particular, there has been long interest in Mersenne prime, which are
closely related to finding perfect numbers and large primes. As of April 2013,
only 48 Mersenne primes were discovered so far. Among them, the largest
known prime ( discovered on January 25th , 2013) is 257885161 − 1, which is
17425170 digits long ; see http://www.mersenne.org/various/57885161.htm and
http://primes.utm.edu/mersenne/ for more details.