Let p = any prime number.
(2p -1) is called a Mersenne number. Any such number that is
prime is called a Mersenne Prime. Father Mersenne wrote a list of
numbers of this type which he thought were prime, but a few were
not.
In fact, most of the large Mersenne numbers are not prime, but
all the really large numbers that have been proved to be prime are
Mersenne Primes.