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 exponent (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, 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 is 2, 3, 5, or 7 — the first four prime numbers!