Mersenne prime

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. For example, 3 = 4 − 1 = 2^{2} − 1 is a Mersenne prime; so is 7 = 8 − 1 = 2^{3} − 1. On the other hand, 15 = 16 − 1 = 2^{4} − 1, for example, is not a prime.
More generally, Mersenne numbers (not necessarily primes, but candidates for primes) are numbers that are one less than a power of two; hence,
 M_{n} = 2^{n} − 1.
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.
Contents 
Properties of Mersenne numbers
Mersenne numbers share several properties:
M_{n} is a sum of binomial coefficients: <math> M_n = \sum_{i=0}^{n} {n \choose i}  1 <math> .
If a is a divisor of M_{q} (q prime) then a has the following properties: :<math> a \equiv 1 \pmod{2q} <math> and: <math> a \equiv \pm 1 \pmod{8} <math> .
A theorem from Euler about numbers of the form 1+6k shows that M_{q} (q prime) is a prime if and only if there exists only one pair <math> (x,y) <math> such that: <math> M_q = {(2x)}^2 + 3{(3y)}^2 <math> with <math> q \geq 5 <math> . More recently, Bas Jansen has studied <math> M_q = x^2 + dy^2 <math> for d=0..48 and has provided a new (and clearer) proof for case d=3 .
Let <math> q = 3 \ \pmod{4} <math> be a prime. <math> 2q+1 <math> is also a prime if and only if <math> 2q+1 <math> divides <math> M_q <math> .
Reix has recently found that prime and composite Mersenne numbers M_{q} (q prime > 3) can be written as: <math> M_q = {(8x)}^2  {(3qy)}^2 = {(1+Sq)}^2  {(Dq)}^2 <math> . Obviously, if there exists only one pair (x,y), then M_{q} is prime.
Ramanujan has showed that the equation: <math> M_q = 6+x^2 <math> has only 3 solutions with q prime: 3, 5, and 7 (and 2 solutions with q composite).
Searching for Mersenne primes
The calculation
 <math>(2^a1)\cdot (1+2^a+2^{2a}+2^{3a}+\dots+2^{(b1)a})=2^{ab}1<math>
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, 2^{11} − 1 = 23 · 89.
Fast algorithms for finding Mersenne primes are available, and this is why the largest known prime numbers today are Mersenne primes.
The first four Mersenne primes M_{2}, M_{3}, M_{5}, M_{7} were known in antiquity. The fifth, M_{13}, was discovered anonymously before 1461; the next two (M_{17} and M_{19}) were found by Cataldi in 1588. After more than a century M_{31} was verified to be prime by Euler in 1750. The next (in historical, not numerical order) was M_{127}, found by Lucas in 1876, then M_{61} by Pervushin in 1883. Two more  M_{89} and M_{107}  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 M_{67} and M_{257}, and omitted M_{61}, M_{89} and M_{107}.
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 LucasLehmer test. Specifically, it can be shown that M_{n} = 2^{n} − 1 is prime if and only if M_{n} evenly divides S_{n2}, where S_{0} = 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, M_{521}, 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 thirtyeight years; the next one, M_{607}, was found by the computer a little less than two hours later. Three more  M_{1279}, M_{2203}, M_{2281}  were found by the same program in the next several months.
As of February 2005, only 42 Mersenne primes were known; the largest known prime number (2^{25,964,951} − 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).
List of Mersenne primes
The table below lists all known Mersenne primes Template:OEIS:
#  n  M_{n}  Digits in M_{n}  Date of discovery  Discoverer 

1  2  3  1  ancient  ancient 
2  3  7  1  ancient  ancient 
3  5  31  2  ancient  ancient 
4  7  127  3  ancient  ancient 
5  13  8191  4  1456  anonymous 
6  17  131071  6  1588  Cataldi 
7  19  524287  6  1588  Cataldi 
8  31  2147483647  10  1772  Euler 
9  61  2305843009213693951  19  1883  Pervushin 
10  89  618970019…449562111  27  1911  Powers 
11  107  162259276…010288127  33  1914  Powers 
12  127  170141183…884105727  39  1876  Lucas 
13  521  686479766…115057151  157  January 30 1952  Robinson 
14  607  531137992…031728127  183  January 30 1952  Robinson 
15  1,279  104079321…168729087  386  June 25 1952  Robinson 
16  2,203  147597991…697771007  664  October 7 1952  Robinson 
17  2,281  446087557…132836351  687  October 9 1952  Robinson 
18  3,217  259117086…909315071  969  September 8 1957  Riesel 
19  4,253  190797007…350484991  1,281  November 3 1961  Hurwitz 
20  4,423  285542542…608580607  1,332  November 3 1961  Hurwitz 
21  9,689  478220278…225754111  2,917  May 11 1963  Gillies 
22  9,941  346088282…789463551  2,993  May 16 1963  Gillies 
23  11,213  281411201…696392191  3,376  June 2 1963  Gillies 
24  19,937  431542479…968041471  6,002  March 4 1971  Tuckerman 
25  21,701  448679166…511882751  6,533  October 30 1978  Noll & Nickel 
26  23,209  402874115…779264511  6,987  February 9 1979  Noll 
27  44,497  854509824…011228671  13,395  April 8 1979  Nelson & Slowinski 
28  86,243  536927995…433438207  25,962  September 25 1982  Slowinski 
29  110,503  521928313…465515007  33,265  January 28 1988  Colquitt & Welsh 
30  132,049  512740276…730061311  39,751  September 20 1983  Slowinski 
31  216,091  746093103…815528447  65,050  September 6 1985  Slowinski 
32  756,839  174135906…544677887  227,832  February 19 1992  Slowinski & Gage 
33  859,433  129498125…500142591  258,716  January 10 1994  Slowinski & Gage 
34  1,257,787  412245773…089366527  378,632  September 3 1996  Slowinski & Gage 
35  1,398,269  814717564…451315711  420,921  November 13 1996  GIMPS / Joel Armengaud 
36  2,976,221  623340076…729201151  895,932  August 24 1997  GIMPS / Gordon Spence 
37  3,021,377  127411683…024694271  909,526  January 27 1998  GIMPS / Roland Clarkson 
38  6,972,593  437075744…924193791  2,098,960  June 1 1999  GIMPS / Nayan Hajratwala 
39^{*}  13,466,917  924947738…256259071  4,053,946  November 14 2001  GIMPS / Michael Cameron 
40^{*}  20,996,011  125976895…855682047  6,320,430  November 17 2003  GIMPS / Michael Shafer 
41^{*}  24,036,583  299410429…733969407  7,235,733  May 15 2004  GIMPS / Josh Findley 
42^{*}  25,964,951  122164630…577077247  7,816,230  February 18 2005  GIMPS / Martin Nowak 
^{*}It is not known whether any undiscovered Mersenne primes exist between the 38th (M_{6972593}) and the 42nd (M_{25964951}) on this chart; the ranking is therefore provisional.
For a list of the first 30 Mersenne primes with all digits written out, see Wikisource:Mersenne primes.
See also
 Fermat prime
 ErdösBorwein constant
 Great Internet Mersenne Prime Search
 New Mersenne conjecture
 Prime95 / MPrime
 LucasLehmer test
 Double Mersenne number
 Mersenne twister
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
 The first 30 Mersenne primes written out in decimal
 GIMPS status page http://www.mersenne.org/status.htm gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 3942
 Discovery of the 42nd (http://mathworld.wolfram.com/news/20050226/mersenne/)
 Mersenne numbers (http://mathworld.wolfram.com/MersenneNumber.html)
 prime Mersenne numbers (http://mathworld.wolfram.com/MersennePrime.html)
 Slashdot  Discovery of the 42nd (http://science.slashdot.org/science/05/02/26/1814202.shtml?tid=228)
 M_{q} = (8x)^2  (3qy)^2 Proof (http://tony.reix.free.fr/Mersenne/Mersenne8x3qy.pdf)
 M_{q} = x^2 + d.y^2 Thesis (http://www.math.leidenuniv.nl/scripties/jansen.ps)da:Mersennetal
de:MersennePrimzahl es:Número primo de Mersenne fr:Nombre premier de Mersenne is:Mersenne frumtölur it:Numero primo di Mersenne he:מספר מרסן lt:Merseno skaičiai nl:Mersennepriemgetal ja:メルセンヌ数 pl:Liczby Mersenne'a ru:Число Мерсенна scn:nummuru primu di Mersenne sl:Mersennovo število sv:Mersenneprimtal zh:梅森素数