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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 Mn = 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. (2ab − 1 is divisible by both 2a − 1 and 2b − 1.) The definition is therefore unchanged when written Mp = 2p − 1 where p is assumed prime.

More generally, numbers of the form Mn = 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 Mp are also noteworthy due to their connection to perfect numbers.

As of January 2016, 49 Mersenne primes are known. The largest known prime number 274,207,281 − 1 is a Mersenne prime.


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