Some definitions of Mersenne numbers require that the exponent n be prime.
A Mersenne prime is a Mersenne number that is prime., only 46 Mersenne primes are known; the largest known prime number (243,112,609 − 1) is a Mersenne prime, and 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 digits.
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is infinite. 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 Mn to be a Mersenne prime, the exponent n 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, it may nevertheless turn out that 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 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.
shows that Mn can be prime only if n itself is prime—that is, the primality of n is necessary but not sufficient for Mn to be prime—which simplifies the search for Mersenne primes considerably. The converse statement, namely that Mn is necessarily prime if n 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 2008 are Mersenne primes.
The first four Mersenne primes , , and were known in antiquity. The fifth, , was discovered anonymously before 1461; the next two ( and ) were found by Cataldi in 1588. After nearly two centuries, was verified to be prime by Euler in 1772. The next (in historical, not numerical order) was , found by Lucas in 1876, then by Pervushin in 1883. Two more ( and ) 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 test for Mersenne numbers. Specifically, it can be shown that (for ) is prime if and only if Mn divides Sn−2, where and for , .
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, 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 appear to have 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 is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23. This is the eighth Mersenne prime discovered at UCLA.
by setting c = 2a, d = 1, and n = b
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.
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.
If q divides 2p − 1 then 2p ≡ 1 (mod q). By Fermat's Little Theorem, 2(q − 1) ≡ 1 (mod q). Assume there exists such a p which does not divide q − 1. Then as p and q − 1 must be relatively prime, a similar application of Fermat's Little Theorem says that (q − 1)(p − 1) ≡ 1 (mod p). Thus there is a number x ≡ (q − 1)(p − 2) for which (q − 1)·x ≡ 1 (mod p), and therefore a number k for which (q − 1)·x − 1 = kp. Since 2(q − 1) ≡ 1 (mod q), raising both sides of the congruence to the power x gives 2(q − 1)x ≡ 1, and since 2p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2kp ≡ 1. Thus 2(q − 1)x ÷ 2kp = 2(q − 1)x − kp ≡ 1 (mod q). But by definition, (q − 1)x − kp = 1, implying that 21 ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p does not divide q − 1 is untenable.
|#||n||Mn||Digits in Mn||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|
|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||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||43,112,609||316470269…697152511||12,978,189||August 23 2008||GIMPS / Edson Smith|
To help visualize the size of the 46th 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.