Some definitions of Mersenne numbers require that the exponent p be prime.
A Mersenne prime is a Mersenne number that is prime. It is known that if 2p − 1 is prime then p is prime so it makes no difference which Mersenne number definition is used. As of October 2009, only 47 Mersenne primes are known; the largest known prime number (243,112,609 − 1) is a Mersenne prime. In modern times, the largest known prime has almost always been a Mersenne prime. Like several previously-discovered Mersenne primes, it was discovered by a distributed computing project on the Internet, known as the Great Internet Mersenne Prime Search (GIMPS). It was the first known prime number with more than 10 million base-10 digits.
|Are there infinitely many Mersenne primes?|
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether 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. It is also 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 Mp to be a Mersenne prime, the exponent p itself must be a prime number. This rules out primality for numbers such as M4 = 24 − 1 = 15: since the exponent 4 = 2×2 is composite, the theorem predicts that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are
While it is true that only Mersenne numbers Mp, where p = 2, 3, 5, … could be prime, often Mp is not prime even for a prime exponent p. The smallest counterexample is the Mersenne number
which is not 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 quickly, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The 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.
shows that Mp can be prime only if p itself is prime—that is, the primality of p is necessary but not sufficient for Mp to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that Mp is necessarily prime if p is prime, is false. The smallest counterexample is 211 − 1 = 2,047 = 23 × 89, a composite number.
Fast algorithms for finding Mersenne primes are available, and the largest known prime numbers as of 2009 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.
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 and improved by Lehmer in the 1930s, now known as the Lucas–Lehmer primality test. Specifically, it can be shown that (for p > 2) Mp = 2p − 1 is prime if and only if Mp divides Sp−2, where S0 = 4 and, for k > 0,
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949. But 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, M44497 is the first gigantic, and M6,972,593 was the first megaprime to be discovered, being a prime with at least 1,000,000 digits. All three were the first known prime of any kind of that size.
In September 2008, mathematicians at UCLA participating in GIMPS won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This is the eighth Mersenne prime discovered at UCLA.
On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. This report was apparently overlooked until June 4, 2009. The find was verified on June 12, 2009. The prime is 242,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is less than the largest known which was the 45th to be discovered.
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 M67 and M257 (which are composite), and omitted M61, M89, and M107 (which are prime). Mersenne gave little indication how he came up with his list, and its rigorous verification was completed more than two centuries later.
|#||p||Mp||Digits in Mp||Date of discovery||Discoverer|
|1||2||3||1||5th century BC||Ancient Greek mathematicians|
|2||3||7||1||5th century BC||Ancient Greek mathematicians|
|3||5||31||2||3rd century BC||Ancient Greek mathematicians|
|4||7||127||3||3rd century BC||Ancient Greek mathematicians|
|13||521||686479766…115057151||157||January 30, 1952||Robinson, using SWAC (computer)|
|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, using BESK|
|19||4,253||190797007…350484991||1,281||November 3, 1961||Hurwitz, using IBM 7090|
|20||4,423||285542542…608580607||1,332||November 3, 1961||Hurwitz|
|21||9,689||478220278…225754111||2,917||May 11, 1963||Gillies, using ILLIAC II|
|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, using IBM 360/91|
|25||21,701||448679166…511882751||6,533||October 30, 1978||Noll & Nickel, using CDC Cyber 174|
|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||Slowinski|
|31||216,091||746093103…815528447||65,050||September 1, 1985||Slowinski|
|32||756,839||174135906…544677887||227,832||February 19, 1992||Slowinski & Gage on Harwell Lab Cray-2|
|33||859,433||129498125…500142591||258,716||January 4, 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|
|43[*]||30,402,457||315416475…652943871||9,152,052||December 15, 2005||GIMPS / Curtis Cooper & Steven Boone|
|44[*]||32,582,657||124575026…053967871||9,808,358||September 4, 2006||GIMPS / Curtis Cooper & Steven Boone|
|45[*]||37,156,667||202254406…308220927||11,185,272||September 6, 2008||GIMPS / Hans-Michael Elvenich|
|46[*]||42,643,801||169873516…562314751||12,837,064||April 12, 2009[**]||GIMPS / Odd M. Strindmo|
|47[*]||43,112,609||316470269…697152511||12,978,189||August 23, 2008||GIMPS / Edson Smith|
* It is not known whether any undiscovered Mersenne primes exist between the 39th (M13,466,917) and the 47th (M43,112,609) on this chart; the ranking is therefore provisional. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, the current record holder was followed by 2 smaller Mersenne primes, first 2 weeks later and then 8 months later.
** M42,643,801 was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad.
To help visualize the size of the 47th known Mersenne prime, it would require 3,461 pages to display the number in base 10 with 75 digits per line and 50 lines per page.
The factorization of a prime number is by definition the number itself. This section is about composite numbers. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm 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 can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor. As of 2010, the composite Mersenne number with largest proven prime factors is 220887 − 1, which is known to have a factor p with 6229 digits that was proven prime with ECPP. The largest with probable prime factors allowed is 2684127 − 1 = 23765203727 × q, where q is a probable prime.
is an even perfect number (which is also the Mpth triangular 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, but it appears unlikely that there is one.
The binary representation of 2p − 1 is the digit 1 repeated p times, for example, 25 − 1 = 111112 in the binary notation. A Mersenne number is therefore a repunit in base 2, and Mersenne primes are the base 2 repunit primes.
The base 2 representation of a Mersenne number shows the factorization pattern for composite exponent. For example:
In computer science, unsigned p-bit integers can be used to express numbers up to Mp.
In the mathematical problem Tower of Hanoi, solving a puzzle with a p-disc tower requires at least Mp steps.
The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is the fifth Mersenne prime. The asteroids with the previous four numbers corresponding to Mersenne primes (3 Juno, 7 Iris, 31 Euphrosyne, 127 Johanna) were already named after others.