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.

2¹ – 1 = 1

2² – 1 = 3

2³ – 1 = 7

2⁴ – 1 = 15

2⁵ – 1 = 31

2⁶ – 1 = 63

2⁷ – 1 = 127

2⁸ – 1 = 255

2⁹ – 1 = 511

2¹⁰ – 1 = 1,023

Every second Mersenne number — every one that uses an even power of 2 (such as 2⁴ – 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, 2⁶⁷ – 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.)