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 = 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.

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 = 2ⁿ − 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

1 Properties of Mersenne numbers 2 Searching for Mersenne primes 2.1 List of Mersenne primes 3 See also 4 External links

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^a-1)\cdot (1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)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, 2¹¹ − 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 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 M₁₀₇.

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 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	Mn	Digits in Mn	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
• Erdös-Borwein constant
• Great Internet Mersenne Prime Search
• New Mersenne conjecture
• Prime95 / MPrime
• Lucas-Lehmer 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 39-42
• Discovery of the 42nd (http://mathworld.wolfram.com/news/2005-02-26/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

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