Mersenne Prime - Perfect Numbers

Perfect Numbers

Mersenne primes M_p are interesting to many for their connection to perfect numbers. In the 4th century BCE, Euclid demonstrated that, whenever 2p−1 is prime, 2p−1(2p−1) is an even perfect number. This number, also expressible as M_p(M_p+1)/2, is the M_pth triangular number and the 2p−1th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers, but it appears unlikely.