Recall that primes of the form 2p − 1 are called Mersenne primes
We do not know whether or
not there are infinitely many
Mersenne primes
and so we do not know whether or not there exist
infinitely many even perfect numbers. Equally mysterious is the question of whether there are any odd
perfect numbers. For a survey of the (few) known results and the (many) open problems concerning
perfect numbers, we refer the reader to