MERSENNE PRIME

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In mathematics, a 'Mersenne number' is a number that is one less than a power of two,
:''M''_''n'' = 2^''n'' − 1.
The central question about these numbers is which of them are also prime numbers, called the 'Mersenne primes', and in modern times the largest known prime has nearly always been a Mersenne prime^[1]. Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers. They are named after 17th century French scholar Marin Mersenne, who compiled a list of Mersenne primes with exponents up to 257. His list was only partially correct, as Mersenne mistakenly included ''M''₆₇ and ''M''₂₅₇ (which are composite), and omitted ''M''₆₁, ''M''₈₉, and ''M''₁₀₇ (which are prime). Mersenne gave no indication how he came up with his list, and its rigorous verification was completed more than two centuries later.
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether there is a largest Mersenne prime, which would mean that the set of Mersenne primes is finite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. Perhaps even more embarrassingly, it is not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.
A basic theorem about Mersenne numbers states that in order for ''M''_''n'' to be a Mersenne prime, the exponent ''n'' itself must be a prime number. This rules out primality for numbers such as ''M''₄ = 2⁴ -1 = 15: since the exponent 4=2×2 is composite, the theorem says that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are
: ''M''₂ = 3, ''M''₃ = 7, ''M''₅ = 31.
Whilst it is true that only Mersenne numbers ''M''_''p'', where ''p'' = 2, 3, 5, … ''could'' be prime, it may nevertheless turn out that ''M''_''p'' is not prime even for a prime exponent ''p''. The smallest counterexample is the Mersenne number
: ''M''₁₁ = 2¹¹ − 1 = 2047 = 23 × 89,
which is not a Mersenne prime, even though 11 is a prime number. The lack of an obvious rule to determine whether a given Mersenne number is prime makes the search for Mersenne primes an interesting task, which becomes difficult very soon, since Mersenne numbers grow very fast. The Lucas–Lehmer test for Mersenne numbers is an efficient primality test that greatly aids this task. Search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.

Contents

Searching for Mersenne primes

Theorems about Mersenne numbers

List of known Mersenne primes

Factorization of Mersenne numbers

Perfect numbers

Generalization

Notes

See also

External links

Searching for Mersenne primes

The identity
:$2^\{ab\}-1=(2^a-1)cdot\; left(1+2^a+2^\{2a\}+2^\{3a\}+dots+2^\{(b-1)a\}$
ight)
shows that ''M_n'' can be prime only if ''n'' itself is prime, which simplifies the search for Mersenne primes considerably. (This follows very simply from the Mersenne property of the sequence of numbers of the form $x^n\; -\; y^n$. This states that $x^a\; -\; y^a|x^b\; -\; y^b$ if and only if ''a''|''b''.) The converse statement, namely that ''M_n'' is necessarily prime if ''n'' is prime, is false. The smallest counterexample is 2¹¹−1 = 23×89, a composite number.
Fast algorithms for finding Mersenne primes are available, and the largest known prime numbers as of 2007 are Mersenne primes.
The first four Mersenne primes $M\_2=3$, $M\_3=7$, $M\_5=31$ and $M\_7=127$ were known in antiquity. The fifth, $M\_\{13\}=8191$, was discovered anonymously before 1461; the next two ($M\_\{17\}$ and $M\_\{19\}$) were found by Cataldi in 1588. After nearly two centuries, $M\_\{31\}$ was verified to be prime by Euler in 1772. 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 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 1856 [1][2] and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test for Mersenne numbers. Specifically, it can be shown that (for $n>2$) $M\_n=2^n-1$ is prime if and only if ''M_n'' divides ''S_n-2'', 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 thirty-eight 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. ''M''₄₂₅₃ is the first Mersenne prime that is titanic, ''M''₄₄₄₉₇ is the first gigantic, and ''M''_6,972,593 is the first megaprime, meaning a prime with at least 1,000,000 digits.[3] All three were the first known prime of any kind of that size.
As of August 2007, only 44 Mersenne primes are known; the largest known prime number (2^32,582,657−1) 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).

Theorems about Mersenne numbers

★ 1) If ''n'' is a positive integer, by the binomial theorem we can write:
:$c^n-d^n=(c-d)sum\_\{k=0\}^\{n-1\}\; c^kd^\{n-1-k\}$,
or
:$(2^a-1)cdot\; left(1+2^a+2^\{2a\}+2^\{3a\}+dots+2^\{(b-1)a\}$
ight)=2^{ab}-1
by setting $c=2^a$, $d=1$, and $n=b$
''proof''
:$(a-b)sum\_\{k=0\}^\{n-1\}a^kb^\{n-1-k\}$
:$=sum\_\{k=0\}^\{n-1\}a^\{k+1\}b^\{n-1-k\}-sum\_\{k=0\}^\{n-1\}a^kb^\{n-k\}$
:$=a^n+sum\_\{k=1\}^\{n-1\}a^kb^\{n-k\}-sum\_\{k=1\}^\{n-1\}a^kb^\{n-k\}-b^n$
:$=a^n-b^n$

★ 2) If $2^n-1$ is prime, then $n$ is prime.
''proof''
By
:$(2^a-1)cdot\; left(1+2^a+2^\{2a\}+2^\{3a\}+dots+2^\{(b-1)a\}$
ight)=2^{ab}-1
If $n$ is not prime, or $n=ab$ where
.
Therefore, $2^a-1$ would divide $2^n-1$,
or $2^n-1$ is not prime.

★ 3) If ''p'' is an odd prime, then any prime ''q'' that divides $2^p-1$ must be ''1'' plus a multiple of ''2p''. This holds of course even when $2^p-1$ is prime. Example I: $2^5-1=31$
is prime, and ''31'' is ''1'' plus a multiple of ''2
★ 5''. Example II: $2^\{11\}-1$=''23
★ 89'',
''23=1+2
★ 11'', and ''89=1+8
★ 11'', and also ''23
★ 89=1+186
★ 11''.
''proof''
If ''q'' divides $2^p-1$ then $2^p$ is congruent to ''1'' mod ''q'', so ''p'' divides the order of the multiplicative group mod ''q'', by Lagrange's Theorem. This group has order ''q-1'', so ''q-1=kp'' for some ''k'', and ''q=1+kp''. But ''q'' must be odd, and ''p'' is odd,(except for ''p=2'') so ''k'' is even.

★ 4) If ''p'' is an odd prime, then any prime ''q'' that divides $2^p-1$ must be $pm\; 1\; pmod\; 8$. Proof: $2^\{p+1\}\; =\; 2\; pmod\; q$, so $2^\{(p+1)/2\}$ is a square root of 2 modulo $q$. By quadratic reciprocity, any prime modulo which two has a square root is $pm\; 1\; pmod\; 8$.

List of known Mersenne primes

The table below lists all known Mersenne primes :

#	''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'' [4]
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[5]
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 19 1983[6]	Slowinski
31	216,091	746093103…815528447	65,050	September 1 1985[7]	Slowinski
32	756,839	174135906…544677887	227,832	February 19 1992	Slowinski & Gage on Harwell Lab Cray-2 [8]
33	859,433	129498125…500142591	258,716	January 4 1994 [9]	Slowinski & Gage
34	1,257,787	412245773…089366527	378,632	September 3 1996	Slowinski & Gage [10]
35	1,398,269	814717564…451315711	420,921	November 13 1996	GIMPS / Joel Armengaud [11]
36	2,976,221	623340076…729201151	895,932	August 24 1997	GIMPS / Gordon Spence [12]
37	3,021,377	127411683…024694271	909,526	January 27 1998	GIMPS / Roland Clarkson [13]
38	6,972,593	437075744…924193791	2,098,960	June 1 1999	GIMPS / Nayan Hajratwala [14]
39	13,466,917	924947738…256259071	4,053,946	November 14 2001	GIMPS / Michael Cameron [15]
40 ★	20,996,011	125976895…855682047	6,320,430	November 17 2003	GIMPS / Michael Shafer [16]
41 ★	24,036,583	299410429…733969407	7,235,733	May 15 2004	GIMPS / Josh Findley [17]
42 ★	25,964,951	122164630…577077247	7,816,230	February 18 2005	GIMPS / Martin Nowak [18]
43 ★	30,402,457	315416475…652943871	9,152,052	December 15 2005	GIMPS / Curtis Cooper & Steven Boone [19]
44 ★	32,582,657	124575026…053967871	9,808,358	September 4 2006	GIMPS / Curtis Cooper & Steven Boone [20]

^★ It is not known whether any undiscovered Mersenne primes exist between the 39th (''M''_13,466,917) and the 44th (''M''_32,582,657) on this chart; the ranking is therefore provisional.
To help visualize the size of the 44th known Mersenne Prime, a standard word processor layout (12pt Times New Roman, 1" margins) would require 2,769 pages to display the number in base 10.

Factorization of Mersenne numbers

Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorised has been a Mersenne number. As of March 2007, $2^\{1039\}-1$ is the record-holder, after a calculation taking about a year on a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland. See integer factorization records for links to more information.

Perfect numbers

Mersenne primes are interesting to many for their connection to perfect numbers. In the 4th century BC, Euclid demonstrated that if ''M''_''p'' is a Mersenne prime then
: 2^''p''-1×(2^''p''-1) = ''M''_''p''(''M''_''p''+1)/2
is an even perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers.

Generalization

The binary representation of 2^''n'' − 1 is the digit 1 repeated ''n'' times, for example, 2⁵ − 1 = 11111₂ in the binary notation. The Mersenne primes are therefore the base-2 repunit primes.

Notes

1. The largest prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "The Largest Known Prime by Year: A Brief History", from ''The Prime Pages'' website, U of Tennessee at Martin.

See also

★ Repunit

★ Fermat prime

★ Erdős–Borwein constant

★ Mersenne conjectures

★ Prime95 / MPrime

★ Lucas–Lehmer test for Mersenne numbers

★ Double Mersenne number

★ Mersenne twister

★ Wieferich prime

External links

★ Great Internet Mersenne Prime Search (GIMPS) Orlando Florida - home page of mersenne.org

★ prime Mersenne Numbers - History, Theorems and Lists Explanation

★ GIMPS Mersenne Prime - status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 40-44

★

★

★ ''M_q = (8x)² - (3qy)²'' Mersenne Proof (pdf)

★ ''M_q = x² + d.y²'' Math Thesis (ps)

★ Mersenne Prime Bibliography with hyperlinks to original publications

★ dpa - reportage about prime Mersenne number - detection in detail (German)

★ Mersenne prime Wiki

★ 44th Mersenne Prime Found article at MathWorld