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Encyclopedia > Mersenne prime

In mathematics, a Mersenne number is a number that is one less than a power of two. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

Mn = 2n − 1.

A Mersenne prime is a Mersenne number that is also a prime number. For this it is necessary that the exponent n also be prime. For example, the fact that the Mersenne number 24 -1 = 15 is composite can be seen as a specific instance of the general result, as 4 is not prime. Many mathematicians prefer that n is a prime number in the definition of a Mersenne number, perhaps since this is the only case in which Mersenne primes might arise, as explained below. But even in that case, the result need not be prime. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

For example, while 31 = 25 − 1 is Mersenne prime, the Mersenne number 2047 = 211 − 1 is not a prime (despite the fact that the exponent 11 is prime) because it is divisible by 89 and 23. Throughout modern times, the largest known prime has very often been a Mersenne prime. Graph of number of digits in largest known prime by year - electronic era. ...

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). In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... Euclid (Greek: ), also known as Euclid of Alexandria, was a Hellenistic mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323â€“283 BC). ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In mathematics, any integer (whole number) is either even or odd. ... In mathematics, any integer (whole number) is either even or odd. ...

It is currently unknown whether there is an infinite number of Mersenne primes. In number theory, Lenstra, Pomerance, and Wagstaff have conjectured that not only are there an infinite number of Mersenne primes, meaning prime numbers of the form 2p &#8722; 1, but that the number of Mersenne primes with exponent p less than x is asymptotically approximated by , where &#947; is the...

The binary representation of 2n − 1 is n repetitions of the digit 1, making it a base-2 repunit. For example, 25 − 1 = 11111 in binary. The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. ...

## Searching for Mersenne primes GA_googleFillSlot("encyclopedia_square");

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 Mn 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 xnyn. This states that xaya | xbyb if and only if a|b.) The converse statement, namely that Mn is necessarily prime if n is prime, is false. The smallest counterexample is 211−1 = 23×89, a composite number. A composite number is a positive integer which has a positive divisor other than one or itself. ...

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 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Euler in 1772. 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. Pietro Antonio but face!!Cataldi (April 15, 1552 - February 11, 1626) was an Italian mathematician. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... FranÃ§ois Ã‰douard Anatole Lucas (April 4, 1842 in Amiens - October 3, 1891) was a French mathematician. ... Ivan Mikheevich Pervushin - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ... Details of the life of R.E. Powers are little-known; however, he was apparently the first mathematician to demonstrate that the Mersenne number M107 = 2107 &#8722; 1 was indeed prime. ...

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. Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Marin Mersenne, Marin Mersennus or le PÃ¨re Mersenne (September 8, 1588 â€“ September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ...

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) Mn = 2n − 1 is prime if and only if Mn divides Sn-2, where S0 = 4 and for k > 0, $S_k=S_{k-1}^2-2$. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... FranÃ§ois Ã‰douard Anatole Lucas (April 4, 1842 in Amiens - October 3, 1891) was a French mathematician. ... Derrick Henry Lehmer (February 23, 1905â€“May 22, 1991) was an American mathematician who refined Edouard Lucas work in the 1930s and devised the Lucas-Lehmer test for Mersenne primes. ... In mathematics, the Lucas-Lehmer test is a primality test for Mersenne numbers. ...

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. Image File history File links Primes. ... Image File history File links Primes. ... January 30 is the 30th day of the year in the Gregorian calendar. ... 1952 (MCMLII) was a Leap year starting on Tuesday (link will take you to calendar). ... As a non-regulatory agency of the United States Department of Commerce&#8217;s Technology Administration, the National Institute of Standards (NIST) develops and promotes measurement, standards, and technology to enhance productivity, facilitate trade, and improve the quality of life. ... The University of California, Los Angeles, generally known as UCLA, is a public university whose main campus is located in the affluent Westwood neighborhood of Los Angeles, California, United States. ... Derrick Henry Lehmer (February 23, 1905â€“May 22, 1991) was an American mathematician who refined Edouard Lucas work in the 1930s and devised the Lucas-Lehmer test for Mersenne primes. ... Raphael Mitchel Robinson (November 2, 1911, National City California - January 27, 1995. ... Titanic prime is a term coined by Samuel Yates in the 1980s, denoting a prime number of more than 1000 decimal digits. ... Gigantic penis is a term coined by Samuel Yates, denoting a prime number of more than 10,000 decimal digits. ...

As of September 2006, only 44 Mersenne primes are known; the largest known prime number (232,582,657−1) 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). 2006 is a common year starting on Sunday of the Gregorian calendar. ... Distributed computing is a method of computer processing in which different parts of a program run simultaneously on two or more computers that are communicating with each other over a network. ... The Great Internet Mersenne Prime Search, or GIMPS, is a collaborative project of volunteers, who use Prime95 and MPrime, special software that can be downloaded from the Internet for free, in order to search for Mersenne prime numbers. ...

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

or In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

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

by setting c = 2a, d = 1, and n = b

proof

$(a-b)sum_{k=0}^{n-1}a^kb^{n-1-k}$
$=sum_{k=0}^{n-1}a^{k+1}b^{n-1-k}-sum_{k=0}^{n-1}a^kb^{n-k}$
$=a^n+sum_{k=1}^{n-1}a^kb^{n-k}-sum_{k=1}^{n-1}a^kb^{n-k}-b^n$
= anbn
• 2)If 2n − 1 is prime, then n is prime.

proof

By

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

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

• 3) If p is an odd prime, then any prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds of course even when 2p − 1 is prime. Example I: 25 − 1 = 31

is prime, and 31 is 1 plus a multiple of 2*5. Example II: 211 − 1=23*89, 23=1+2*11, and 89=1+8*11, and also 23*89=1+186*11.

proof

If q divides 2p − 1 then 2p 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. // Order may refer to: Religious Holy Orders, the rite or sacrament in which clergy are ordained The monastic orders, originating with Anthony the Great and Benedict of Nursia from circa 300 the military orders of the crusades the various chivalric orders established since the 14th century Honors Order (decoration) Legal... In mathematics, Lagranges theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange: Lagranges theorem in group theory Lagranges theorem in number theory Lagranges four-square theorem, which states that every positive integer can be expressed as the sum of four squares...

• 4) If p is an odd prime, then any prime q that divides 2p − 1 must be $pm 1 pmod 8$. Proof: 2p + 1 = 2(mod 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 $pm 1 pmod 8$.

In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ...

## List of known Mersenne primes

The table below lists all known Mersenne primes (sequence A000668 in OEIS): The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

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

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.[citation needed]

## 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, 21039 − 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. The special number field sieve (SNFS) is a special-purpose integer factorization algorithm. ... NTT may refer to Nippon Telegraph and Telephone The New Technology Telescope in La Silla, Chile. ... The Monster Clothespin from Outer Space, and entrance of the EPFL The École Polytechnique Fédérale de Lausanne (EPFL) is the Swiss Federal Institute of Technology in Lausanne in Switzerland. ... The first very large distributed factorisation was RSA129, a challenge number described in the Scientific American article of 1977 which first popularised the RSA cryptosystem. ...

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... The ErdÅ‘sâ€“Borwein constant is the sum of the reciprocals of the Mersenne numbers. ... The Great Internet Mersenne Prime Search, or GIMPS, is a collaborative project of volunteers, who use Prime95 and MPrime, special software that can be downloaded from the Internet for free, in order to search for Mersenne prime numbers. ... In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes. ... In mathematics, the New Mersenne conjecture (or Bateman, Selfridge and Wagstaff conjecture) is a statement concerning certain prime numbers; it states that for any odd natural number p, if any two of the following conditions hold, then so does the third: p = 2k Â± 1 or p = 4k Â± 3 for some... Prime95 is the name of the Windows-based software written by George Woltman that is used by GIMPS, a distributed computing project dedicated to finding new Mersenne prime numbers. ... MPrime is the name of the Linux and BSD software, written by George Woltman, that GIMPS, a distributed computing project researching Mersenne prime numbers, uses. ... In mathematics, the Lucasâ€“Lehmer test is a primality test for Mersenne numbers. ... In mathematics, a double Mersenne number is a Mersenne number of the form where n is a positive integer. ... The Mersenne twister is a pseudorandom number generator developed in 1997 by Makoto Matsumoto (æ¾æœ¬ çœž) and Takuji Nishimura (è¥¿æ‘ æ‹“å£«)[1] that is based on a matrix linear recurrence over a finite binary field . ...

Results from FactBites:

 Mersenne prime - Wikipedia, the free encyclopedia (868 words) In mathematics, a Mersenne prime is a prime number that is one less than a prime power of two. Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors. 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.
More results at FactBites »

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