Mersenne primes are named after a 17th-century French monk and mathematician, Marin Mersenne.

A Mersenne number is a number of the form 2^p - 1, where p is a prime.

A Mersenne prime is a prime number of the form 2^p - 1.

Theorem. If 2^p - 1 is a Mersenne prime, then p is prime.

Mersenne Primes.
3 = 2²-1,    7 = 2³-1,    31 = 2⁵-1,    127 = 2⁷-1.
2047 = 2¹¹-1 = 23 x 89 is not a Mersenne prime.

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding itself.

Perfect Numbers.    6 = 1+2+3,    28 = 1+2+4+7+14,    496,    8128.

Theorem. n is an even perfect number iff n = 2^p-1 (2^p - 1)
for some prime p, and 2^p - 1 is also prime.