Mersenne primes

Mersenne prime

Named after	Marin Mersenne
Publication year	1536
Author of publication	Regius, H.
No. of known terms	49
Conjectured no. of terms	Infinite
Subsequence of	Mersenne numbers
First terms	3, 7, 31, 127
Largest known term	274,207,281 − 1 (January 2016)
OEIS index	A000668

2	3	5	7	11	13	17	19
23	29	31	37	41	43	47	53
59	61	67	71	73	79	83	89
97	101	103	107	109	113	127	131
137	139	149	151	157	163	167	173
179	181	191	193	197	199	211	223
227	229	233	239	241	251	257	263
269	271	277	281	283	293	307	311
The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold.

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number that can be written in the form M_n = 2ⁿ − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes (sequence in the OEIS) are 3, 7, 31, and 127.

If n is a composite number then so is 2ⁿ − 1. (2^ab − 1 is divisible by both 2^a − 1 and 2^b − 1.) The definition is therefore unchanged when written M_p = 2^p − 1 where p is assumed prime.

More generally, numbers of the form M_n = 2ⁿ − 1 without the primality requirement are called Mersenne numbers. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those numbers whose binary representation contains a prime number of ones and no zeros. The smallest composite pernicious Mersenne number is 2¹¹ − 1 = 2047 = 23 × 89.

Mersenne primes M_p are also noteworthy due to their connection to perfect numbers.

As of January 2016^[ref], 49 Mersenne primes are known. The largest known prime number 2^74,207,281 − 1 is a Mersenne prime.

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