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 = 2ⁿ − 1.

A Mersenne prime is a Mersenne number that is a prime number. It is necessary for n to be prime for 2^n-1 to be prime, but the converse is not true. Many mathematicians prefer the definition that n has to be a prime number.

For example, 31 = 2⁵ − 1, and 5 is a prime number, so 31 is a Mersenne number; and 31 is also a Mersenne prime because it is a prime number. But the Mersenne number 2047 = 2¹¹ − 1 is not a prime because it is divisible by 89 and 23. And 2⁴ -1 = 15 can be shown to be composite because 4 is not prime.

Throughout modern times, the largest known prime number has very often been a Mersenne prime. Most sources restrict the term Mersenne number to where n is prime, as all Mersenne primes must be of this form as seen below.

Mersenne primes have a close connection to perfect numbers, which are numbers 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. In the 18th century, Leonhard Euler proved that all even perfect numbers have this form. No odd perfect numbers are known, and it is suspected that none exist (any that do have to belong to a significant number of special forms).

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

1 Searching for Mersenne primes
2 Theorems about Mersenne prime
3 List of known Mersenne primes
4 See also
5 External links

[edit] Searching for Mersenne primes

The identity

$2^{ab}-1=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)$

shows that M_n can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. 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 this is why the largest known prime numbers today 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 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. His list was not correct, 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 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 $k > 0$ , $S_k=S_{k-1}^2-2$ .

Graph of number of digits in largest known Mersenne prime by year - electronic era. Note that the vertical scale is logarithmic.

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. M₄₂₅₃ is the first Mersenne prime that is titanic, and M₄₄₄₉₇ is the first gigantic.

As of September 2006, 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).

[edit] Theorems about Mersenne prime

If n is a positive integer, by the Binomial theorem we can write:

$c^n-d^n=(c-d)\sum_{k=0}^{n-1} c^kd^{n-1-k}$ ,

$(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)=2^{ab}-1$

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

If $2 n - 1$ is prime, then $n$ is prime.

proof

$(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\dots+2^{(b-1)a}\right)=2^{ab}-1$

If $n$ is not prime, or $n = a b$ where $1 < a, b < n$ . Therefore, $2 a - 1$ would divide $2 n - 1$ , or $2 n - 1$ is not prime.

[edit] List of known Mersenne primes

The table below lists all known Mersenne primes (sequence A000668 in 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	1750	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 on Harwell Lab Cray-2 [1]
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 [2]
35	1,398,269	814717564…451315711	420,921	November 13, 1996	GIMPS / Joel Armengaud [3]
36	2,976,221	623340076…729201151	895,932	August 24, 1997	GIMPS / Gordon Spence [4]
37	3,021,377	127411683…024694271	909,526	January 27, 1998	GIMPS / Roland Clarkson [5]
38	6,972,593	437075744…924193791	2,098,960	June 1, 1999	GIMPS / Nayan Hajratwala [6]
39	13,466,917	924947738…256259071	4,053,946	November 14, 2001	GIMPS / Michael Cameron [7]
40^*	20,996,011	125976895…855682047	6,320,430	November 17, 2003	GIMPS / Michael Shafer [8]
41^*	24,036,583	299410429…733969407	7,235,733	May 15, 2004	GIMPS / Josh Findley [9]
42^*	25,964,951	122164630…577077247	7,816,230	February 18, 2005	GIMPS / Martin Nowak [10]
43^*	30,402,457	315416475…652943871	9,152,052	December 15, 2005	GIMPS / Curtis Cooper & Steven Boone [11]
44^*	32,582,657	124575026…053967871	9,808,358	September 4, 2006	GIMPS / Curtis Cooper & Steven Boone [12]

Wikinews has news related to:

Distributed computing discovers largest known prime number

Wikinews has news related to:

CMSU computing team discovers another record size prime

^*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.

[edit] See also

[edit] 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
Mersenne numbers - Wolfram Research/Mathematica
prime Mersenne numbers - Wolfram Research/Mathematica
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
43rd Mersenne Prime Found article at MathWorld

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Mersenne prime

From Wikipedia, the free encyclopedia

Contents

[edit] Searching for Mersenne primes

[edit] Theorems about Mersenne prime

[edit] List of known Mersenne primes

[edit] See also

[edit] External links

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