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 M_n = 2ⁿ − 1 is used. The calculation

shows that M_n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. But the converse is not true; M_n 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 M_n = 2ⁿ − 1 is prime if and only if M_n evenly divides S_n-2, where S₀ = 4 and for k > 0, S_k = S_{k − 1}² − 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 (2^20,996,011 − 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):

#	p	Digits in Mp	Date of discovery	Discoverer
1	2	1	ancient	ancient
2	3	1	ancient	ancient
3	5	2	ancient	ancient
4	7	3	ancient	ancient
5	13	4	1456	anonymous
6	17	6	1588	Cataldi
7	19	6	1588	Cataldi
8	31	10	1772	Euler
9	61	19	1883	Pervushin
10	89	27	1911	Powers
11	107	33	1914	Powers
12	127	39	1876	Lucas
13	521	157	1952	Robinson
14	607	183	1952	Robinson
15	1,279	386	1952	Robinson
16	2,203	664	1952	Robinson
17	2,281	687	1952	Robinson
18	3,217	969	1957	Riesel
19	4,253	1,281	1961	Hurwitz
20	4,423	1,332	1961	Hurwitz
21	9,689	2,917	1963	Gillies
22	9,941	2,993	1963	Gillies
23	11,213	3,376	1963	Gillies
24	19,937	6,002	1971	Tuckerman
25	21,701	6,533	1978	Noll & Nickel
26	23,209	6,987	1979	Noll
27	44,497	13,395	1979	Nelson & Slowinski
28	86,243	25,962	1982	Slowinski
29	110,503	33,265	1988	Colquitt & Welsh
30	132,049	39,751	1983	Slowinski
31	216,091	65,050	1985	Slowinski
32	756,839	227,832	1992	Slowinski & Gage
33	859,433	258,716	1994	Slowinski & Gage
34	1,257,787	378,632	1996	Slowinski & Gage
35	1,398,269	420,921	November 13 1996	GIMPS / Joel Armengaud
36	2,976,221	895,932	August 24 1997	GIMPS / Gordon Spence
37	3,021,377	909,526	January 27 1998	GIMPS / Roland Clarkson
38	6,972,593	2,098,960	June 1 1999	GIMPS / Nayan Hajratwala
39?	13,466,917	4,053,946	November 14 2001	GIMPS / Michael Cameron (Canada)
40?	20,996,011	6,320,430	November 17 2003	GIMPS / Michael Shafer

See also

• Fermat prime
• Erdös-Borwein constant

External links

• Mersenne prime section of the Prime Pages: http://www.utm.edu/research/primes/mersenne.shtml
• Mersenne Prime Search home page: http://www.mersenne.org