Small Mersenne Primes
Here are the first ten Mersenne numbers, starting
when n = 1. The ones
that are prime — Mersenne primes — are marked in red.
2^{1} – 1 = 1
2^{2} – 1 = 3
2^{3} – 1 = 7
2^{4} – 1 = 15
2^{5} – 1 = 31
2^{6} – 1 = 63
2^{7} – 1 = 127
2^{8} – 1 = 255
2^{9} – 1 = 511
2^{10} – 1 = 1,023
Every second Mersenne number — every one that
uses an even power of 2 (such as 2^{4} – 1) — is a multiple of 3. Many of the other
Mersenne numbers are composite; for example, 511 = 7 x 73. So although all Mersenne numbers are odd numbers, far less than half of
them are prime.
In fact, mathematicians have proven that a Mersenne
number can only be a prime if the exponent is prime. Note that in the list above, Mersenne primes
occur when the exponent (the power to which 2 is raised) is 2, 3, 5, or 7 — the first four prime
numbers!
(NOTE: This doesn't mean that every
Mersenne number with a prime exponent is prime. For example, 2^{67} – 1 is composite, as shown by a mathematician named Frank Nelson
Cole in 1903. To hear an entertaining story about Cole's demonstration, listen to episode 450 of the radio
show This American Life; find that episode here.)
