Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. The numbers p corresponding to Mersenne primes must themselves be prime, although not all primes p lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. Meanwhile, perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.
There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers. This is due to the Euclid–Euler theorem, partially proved by Euclid and completed by Leonhard Euler: even numbers are perfect if and only if they can be expressed in the form 2p − 1 × (2p − 1), where 2p − 1 is a Mersenne prime. In other words, all numbers that fit that expression are perfect, while all even perfect numbers fit that form. For instance, in the case of p = 2, 22 − 1 = 3 is prime, and 22 − 1 × (22 − 1) = 2 × 3 = 6 is perfect.
It is currently an open problem as to whether there are an infinite number of Mersenne primes and even perfect numbers. x is (eγ / log 2) × log log x, where e is Euler's number, γ is Euler's constant, and log is the natural logarithm. It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a lower bound of 101500.The frequency of Mersenne primes is the subject of the Lenstra–Pomerance–Wagstaff conjecture, which states that the expected number of Mersenne primes less than some given
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2022 [update] , there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. New Mersenne primes are found using the Lucas-Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers.
The displayed ranks are among indices currently known as of 2022 [update] ; while unlikely, ranks may change if smaller ones are discovered. According to GIMPS, all possibilities less than the 48th working exponent p = 57,885,161 have been checked and verified as of October 2021 [update] . The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last 6 digits of each number are shown.
|Rank||p||Mersenne prime||Mersenne prime digits||Perfect number||Perfect number digits||Discovery||Discoverer||Method||Ref.|
|1||2||3||1||6||1||Ancient times||Known to Ancient Greek mathematicians||Unrecorded|
|5||13||8191||4||33550336||8||c. 1456||Anonymous||Trial division|
|8||31||2147483647||10||230584...952128||19||1772||Leonhard Euler||Trial division with modular restrictions|
|9||61||230584...693951||19||265845...842176||37||November 1883||Ivan M. Pervushin||Lucas sequences|
|10||89||618970...562111||27||191561...169216||54||June 1911||Ralph Ernest Powers|
|11||107||162259...288127||33||131640...728128||65||June 1, 1914|
|12||127||170141...105727||39||144740...152128||77||January 10, 1876||Édouard Lucas|
|13||521||686479...057151||157||235627...646976||314||January 30, 1952||Raphael M. Robinson||LLT on SWAC|
|15||1,279||104079...729087||386||541625...291328||770||June 25, 1952|
|16||2,203||147597...771007||664||108925...782528||1,327||October 7, 1952|
|17||2,281||446087...836351||687||994970...915776||1,373||October 9, 1952|
|18||3,217||259117...315071||969||335708...525056||1,937||September 8, 1957||Hans Riesel||LLT on BESK|
|19||4,253||190797...484991||1,281||182017...377536||2,561||November 3, 1961||Alexander Hurwitz||LLT on IBM 7090|
|21||9,689||478220...754111||2,917||114347...577216||5,834||May 11, 1963||Donald B. Gillies||LLT on ILLIAC II|
|22||9,941||346088...463551||2,993||598885...496576||5,985||May 16, 1963|
|23||11,213||281411...392191||3,376||395961...086336||6,751||June 2, 1963|
|24||19,937||431542...041471||6,002||931144...942656||12,003||March 4, 1971||Bryant Tuckerman||LLT on IBM 360/91|
|25||21,701||448679...882751||6,533||100656...605376||13,066||October 30, 1978||Landon Curt Noll & Laura Nickel||LLT on CDC Cyber 174|
|26||23,209||402874...264511||6,987||811537...666816||13,973||February 9, 1979||Landon Curt Noll|
|27||44,497||854509...228671||13,395||365093...827456||26,790||April 8, 1979||Harry L. Nelson & David Slowinski||LLT on Cray-1|
|28||86,243||536927...438207||25,962||144145...406528||51,924||September 25, 1982||David Slowinski|
|29||110,503||521928...515007||33,265||136204...862528||66,530||January 29, 1988||Walter Colquitt & Luke Welsh||LLT on NEC SX-2|
|30||132,049||512740...061311||39,751||131451...550016||79,502||September 19, 1983||David Slowinski et al. (Cray)||LLT on Cray X-MP|
|31||216,091||746093...528447||65,050||278327...880128||130,100||September 1, 1985||LLT on Cray X-MP/24|
|32||756,839||174135...677887||227,832||151616...731328||455,663||February 17, 1992||LLT on Harwell Lab's Cray-2|
|33||859,433||129498...142591||258,716||838488...167936||517,430||January 4, 1994||LLT on Cray C90|
|34||1,257,787||412245...366527||378,632||849732...704128||757,263||September 3, 1996||LLT on Cray T94|
|35||1,398,269||814717...315711||420,921||331882...375616||841,842||November 13, 1996||GIMPS / Joel Armengaud||LLT / Prime95 on 90 MHz Pentium PC|
|36||2,976,221||623340...201151||895,932||194276...462976||1,791,864||August 24, 1997||GIMPS / Gordon Spence||LLT / Prime95 on 100 MHz Pentium PC|
|37||3,021,377||127411...694271||909,526||811686...457856||1,819,050||January 27, 1998||GIMPS / Roland Clarkson||LLT / Prime95 on 200 MHz Pentium PC|
|38||6,972,593||437075...193791||2,098,960||955176...572736||4,197,919||June 1, 1999||GIMPS / Nayan Hajratwala||LLT / Prime95 on IBM Aptiva with 350 MHz Pentium II processor|
|39||13,466,917||924947...259071||4,053,946||427764...021056||8,107,892||November 14, 2001||GIMPS / Michael Cameron||LLT / Prime95 on PC with 800 MHz Athlon T-Bird processor|
|40||20,996,011||125976...682047||6,320,430||793508...896128||12,640,858||November 17, 2003||GIMPS / Michael Shafer||LLT / Prime95 on Dell Dimension PC with 2 GHz Pentium 4 processor|
|41||24,036,583||299410...969407||7,235,733||448233...950528||14,471,465||May 15, 2004||GIMPS / Josh Findley||LLT / Prime95 on PC with 2.4 GHz Pentium 4 processor|
|42||25,964,951||122164...077247||7,816,230||746209...088128||15,632,458||February 18, 2005||GIMPS / Martin Nowak|
|43||30,402,457||315416...943871||9,152,052||497437...704256||18,304,103||December 15, 2005||GIMPS / Curtis Cooper & Steven Boone||LLT / Prime95 on PC at University of Central Missouri|
|44||32,582,657||124575...967871||9,808,358||775946...120256||19,616,714||September 4, 2006|
|45||37,156,667||202254...220927||11,185,272||204534...480128||22,370,543||September 6, 2008||GIMPS / Hans-Michael Elvenich||LLT / Prime95 on PC|
|46||42,643,801||169873...314751||12,837,064||144285...253376||25,674,127||June 4, 2009||GIMPS / Odd Magnar Strindmo||LLT / Prime95 on PC with 3 GHz Intel Core 2 processor|
|47||43,112,609||316470...152511||12,978,189||500767...378816||25,956,377||August 23, 2008||GIMPS / Edson Smith||LLT / Prime95 on Dell OptiPlex PC with Intel Core 2 Duo E6600 processor|
|48||57,885,161||581887...285951||17,425,170||169296...130176||34,850,340||January 25, 2013||GIMPS / Curtis Cooper||LLT / Prime95 on PC at University of Central Missouri|
|*||59,451,331||Lowest unverified milestone|
|49||74,207,281||300376...436351||22,338,618||451129...315776||44,677,235||January 7, 2016||GIMPS / Curtis Cooper||LLT / Prime95 on PC with Intel Core i7-4790 processor|
|50||77,232,917||467333...179071||23,249,425||109200...301056||46,498,850||December 26, 2017||GIMPS / Jonathan Pace||LLT / Prime95 on PC with Intel Core i5-6600 processor|
|51||82,589,933||148894...902591||24,862,048||110847...207936||49,724,095||December 7, 2018||GIMPS / Patrick Laroche||LLT / Prime95 on PC with Intel Core i5-4590T processor|
|*||107,148,487||Lowest untested milestone|
Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(a)=b and σ(b)=a, where σ(n) is equal to the sum of positive divisors of n.
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number.
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.
In number theory, an odd integer n is called an Euler–Jacobi probable prime to base a, if a and n are coprime, and
In number theory, a Woodall number (Wn) is any natural number of the form
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.
A Wilson prime, named after English mathematician John Wilson, is a prime number p such that p2 divides (p − 1)! + 1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p − 1)! + 1.
Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability.
In mathematics, a harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are:
The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes.
Richard Peirce Brent is an Australian mathematician and computer scientist. He is an emeritus professor at the Australian National University. From March 2005 to March 2010 he was a Federation Fellow at the Australian National University. His research interests include number theory, random number generators, computer architecture, and analysis of algorithms.
John Lewis Selfridge, was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.
The largest known prime number is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.
The SWAC was an early electronic digital computer built in 1950 by the U.S. National Bureau of Standards (NBS) in Los Angeles, California. It was designed by Harry Huskey.
Curtis Niles Cooper is an American mathematician. He currently is a professor at the University of Central Missouri, in the Department of Mathematics and Computer Science.
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N = k ⋅ 2n − 1, with k < 2n. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form. For numbers of the form N = k ⋅ 2n + 1, either application of Proth's theorem or one of the deterministic proofs described in Brillhart-Lehmer-Selfridge 1975 are used.
In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted Lenstra–Pomerance–Wagstaff conjecture.