Mersenne number
A Mersenne number is one less than a power of 2, such as 3, 7, 15, 31, 63, and 127.
A Mersenne prime is a prime number that is one less than a power of two, such as 3 (one less than 2x2=4) and 7 (one less than 2x2x2=8). Incredibly, a Mersenne prime results only from a prime number as the power of 2:
If 2p−1 is prime, then p is prime.[1]
As explained by Wolfram MathWorld:[2]
The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (OEIS A000668) corresponding to indices n=2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS A000043). Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number.
A Mersenne number is named after Marin Mersenne, a devoutly religious friar of the Roman Catholic Church Order of Minims order, who focused on these particular prime numbers in the early 1600s.