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# Mersenne prime

A Mersenne prime is a prime number that is one less than a power of two. For example, 3 = 4 − 1 = 22 − 1 is a Mersenne prime; so is 7 = 8 − 1 = 23 − 1. On the other hand, 15 = 16 − 1 = 24 − 1, for example, is not a prime. Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.

Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number. Two millennia later, in the 18th century, Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist.

It is currently unknown whether there is an infinite number of Mersenne primes.

More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than an odd power of two; the notation Mn = 2n − 1 is used. The calculation

shows that Mn can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; Mn may be composite even though n is prime. For example, 211 − 1 = 23 · 89.

The first four Mersenne primes M2, M3, M5, M7 were known in antiquity. The fifth, M13, was discovered anonymously before 1461; the next two (M17 and M19) were found by Cataldi in 1588. After more than a century M31 was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was M127, found by Lucas in 1876, then M61 by Pervushin in 1883. Two more - M89 and M107 - were found early in the 20th century, by Powers in 1911 and 1914, respectively.

The numbers are named after 17th century French mathematician Marin Mersenne, who provided a list of Mersenne primes with exponents up to 257; unfortunately, his list was not correct, though, as he mistakenly included M67 and M257, and omitted M61, M89 and M109.

The best method presently known for testing the primality of Mersenne numbers is based on the computation of a recurring sequence, as developed originally by Lucas in 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test. Specifically, it can be shown that Mn = 2n − 1 is prime if and only if Mn evenly divides Sn-2, where S0 = 4 and for k > 0, Sk = Sk − 12 − 2.

The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. The first successful identification of a Mersenne prime, M521, by this means was achieved at 10:00 P.M. on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of Lehmer, with a computer search program written and run by Prof. R.M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M607, was found by the computer a little less than two hours later. Three more - M1279, M2203, M2281 - were found by the same program in the next several months.

As of December 2003, only 40 Mersenne primes were known; the largest known prime number (220,996,011 − 1) is a Mersenne prime. Like several previous Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS). The table below lists all known Mersenne primes (also confer Sloane's A000043):

#pDigits in MpDate of discoveryDiscoverer
121ancientancient
231ancientancient
352ancientancient
473ancientancient
51341456anonymous
61761588Cataldi
71961588Cataldi
831101772Euler
961191883Pervushin
1089271911Powers
11107331914Powers
12127391876Lucas
135211571952Robinson
146071831952Robinson
151,2793861952Robinson
162,2036641952Robinson
172,2816871952Robinson
183,2179691957Riesel
194,2531,2811961Hurwitz
204,4231,3321961Hurwitz
219,6892,9171963Gillies
229,9412,9931963Gillies
2311,2133,3761963Gillies
2419,9376,0021971Tuckerman
2521,7016,5331978Noll & Nickel
2623,2096,9871979Noll
2744,49713,3951979Nelson & Slowinski
2886,24325,9621982Slowinski
29110,50333,2651988Colquitt & Welsh
30132,04939,7511983Slowinski
31216,09165,0501985Slowinski
32756,839227,8321992Slowinski & Gage
33859,433258,7161994Slowinski & Gage
341,257,787378,6321996Slowinski & Gage
351,398,269420,921November 13 1996GIMPS / Joel Armengaud
362,976,221895,932August 24 1997GIMPS / Gordon Spence
373,021,377909,526January 27 1998GIMPS / Roland Clarkson
386,972,5932,098,960June 1 1999GIMPS / Nayan Hajratwala
39?13,466,9174,053,946November 14 2001GIMPS / Michael Cameron (Canada)
40?20,996,0116,320,430November 17 2003GIMPS / Michael Shafer