The largest known prime number (as of January 2022^{[update]}) is 2^{82,589,933} − 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.^{[1]}
A prime number is a positive integer, excluding 1, with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilise a specialised primality test that is faster than the general one. As of December 2020^{[update]}, the eight largest known primes are Mersenne primes.^{[2]} The last seventeen record primes were Mersenne primes.^{[3]}^{[4]} The binary representation of any Mersenne prime is composed of all 1's, since the binary form of 2^{k} − 1 is simply k 1's.^{[5]}
The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is very fast compared to other known primality tests for other kinds of numbers. With current computers, a multi-million digit Mersenne-like number can be proven prime, but only multi-thousand digit other numbers can be proven prime. Probable primes, such as the base-10 repunit R_{8177207}, pass probabilistic primality tests but are not truly proven prime.
The record is currently held by 2^{82,589,933} − 1 with 24,862,048 digits, found by GIMPS in December 2018.^{[1]} The first and last 120 digits of its value are shown below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...
(24,861,808 digits omitted)
... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591^{[6]}
The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3,000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits.
There are several prizes offered by the Electronic Frontier Foundation for record primes.^{[7]} GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and larger and will split the Electronic Frontier Foundation's US$150,000 prize with a winning participant.
The record passed one million digits in 1999, earning a US$50,000 prize.^{[8]} In 2008, the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.^{[7]} Time called it the 29th top invention of 2008.^{[9]} Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS. Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits.^{[7]}
The following table lists the progression of the largest known prime number in ascending order.^{[3]} Here M_{p} = 2^{p} − 1 is the Mersenne number with exponent p. The longest record-holder known was M_{19} = 524,287, which was the largest known prime for 144 years. No records are known before 1456.
Number | Decimal expansion (only for numbers < M_{1000}) |
Digits | Year found | Discoverer |
---|---|---|---|---|
M_{13} | 8,191 | 4 | 1456 | Anonymous |
M_{17} | 131,071 | 6 | 1588 | Pietro Cataldi |
M_{19} | 524,287 | 6 | 1588 | Pietro Cataldi |
6,700,417 | 7 | 1732 | Leonhard Euler? Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 2^{32} + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.^{[10]} | |
M_{31} | 2,147,483,647 | 10 | 1772 | Leonhard Euler |
999,999,000,001 | 12 | 1851 | Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record. | |
67,280,421,310,721 | 14 | 1855 | Thomas Clausen (but no proof was provided). | |
M_{127} | 170,141,183,460,469, |
39 | 1876 | Édouard Lucas |
20,988,936,657,440, |
44 | 1951 | Aimé Ferrier with a mechanical calculator; the largest record not set by computer. | |
180×(M_{127})^{2}+1 |
521064401567922879406069432539 |
79 | 1951 | J. C. P. Miller & D. J. Wheeler^{[11]} Using Cambridge's EDSAC computer |
M_{521} |
686479766013060971498190079908 |
157 | 1952 | |
M_{607} |
531137992816767098689588206552 |
183 | 1952 | |
M_{1279} | 104079321946...703168729087 | 386 | 1952 | |
M_{2203} | 147597991521...686697771007 | 664 | 1952 | |
M_{2281} | 446087557183...418132836351 | 687 | 1952 | |
M_{3217} | 259117086013...362909315071 | 969 | 1957 | |
M_{4423} | 285542542228...902608580607 | 1,332 | 1961 | |
M_{9689} | 478220278805...826225754111 | 2,917 | 1963 | |
M_{9941} | 346088282490...883789463551 | 2,993 | 1963 | |
M_{11213} | 281411201369...087696392191 | 3,376 | 1963 | |
M_{19937} | 431542479738...030968041471 | 6,002 | 1971 | Bryant Tuckerman |
M_{21701} | 448679166119...353511882751 | 6,533 | 1978 | Laura A. Nickel and Landon Curt Noll^{[12]} |
M_{23209} | 402874115778...523779264511 | 6,987 | 1979 | Landon Curt Noll^{[12]} |
M_{44497} | 854509824303...961011228671 | 13,395 | 1979 | David Slowinski and Harry L. Nelson^{[12]} |
M_{86243} | 536927995502...709433438207 | 25,962 | 1982 | David Slowinski^{[12]} |
M_{132049} | 512740276269...455730061311 | 39,751 | 1983 | David Slowinski^{[12]} |
M_{216091} | 746093103064...103815528447 | 65,050 | 1985 | David Slowinski^{[12]} |
148140632376...836387377151 | 65,087 | 1989 | A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.^{[13]}^{[14]} Largest non-Mersenne prime that was the largest known prime when it was discovered. | |
M_{756839} | 174135906820...328544677887 | 227,832 | 1992 | David Slowinski and Paul Gage^{[12]} |
M_{859433} | 129498125604...243500142591 | 258,716 | 1994 | David Slowinski and Paul Gage^{[12]} |
M_{1257787} | 412245773621...976089366527 | 378,632 | 1996 | David Slowinski and Paul Gage^{[12]} |
M_{1398269} | 814717564412...868451315711 | 420,921 | 1996 | GIMPS, Joel Armengaud |
M_{2976221} | 623340076248...743729201151 | 895,932 | 1997 | GIMPS, Gordon Spence |
M_{3021377} | 127411683030...973024694271 | 909,526 | 1998 | GIMPS, Roland Clarkson |
M_{6972593} | 437075744127...142924193791 | 2,098,960 | 1999 | GIMPS, Nayan Hajratwala |
M_{13466917} | 924947738006...470256259071 | 4,053,946 | 2001 | GIMPS, Michael Cameron |
M_{20996011} | 125976895450...762855682047 | 6,320,430 | 2003 | GIMPS, Michael Shafer |
M_{24036583} | 299410429404...882733969407 | 7,235,733 | 2004 | GIMPS, Josh Findley |
M_{25964951} | 122164630061...280577077247 | 7,816,230 | 2005 | GIMPS, Martin Nowak |
M_{30402457} | 315416475618...411652943871 | 9,152,052 | 2005 | GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone |
M_{32582657} | 124575026015...154053967871 | 9,808,358 | 2006 | GIMPS, Curtis Cooper and Steven Boone |
M_{43112609} | 316470269330...166697152511 | 12,978,189 | 2008 | GIMPS, Edson Smith |
M_{57885161} | 581887266232...071724285951 | 17,425,170 | 2013 | GIMPS, Curtis Cooper |
M_{74207281} | 300376418084...391086436351 | 22,338,618 | 2016 | GIMPS, Curtis Cooper |
M_{77232917} | 467333183359...069762179071 | 23,249,425 | 2017 | GIMPS, Jonathan Pace |
M_{82589933} | 148894445742...325217902591 | 24,862,048 | 2018 | GIMPS, Patrick Laroche |
GIMPS found the fifteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.
A list of the 5,000 largest known primes is maintained by Chris K. Caldwell,^{[15]}^{[16]} of which the twenty largest are listed below.
Rank | Number | Discovered | Digits | Form | Ref |
---|---|---|---|---|---|
1 | 2^{82589933} − 1 | 2018-12-07 | 24,862,048 | Mersenne | ^{[1]} |
2 | 2^{77232917} − 1 | 2017-12-26 | 23,249,425 | Mersenne | ^{[17]} |
3 | 2^{74207281} − 1 | 2016-01-07 | 22,338,618 | Mersenne | ^{[18]} |
4 | 2^{57885161} − 1 | 2013-01-25 | 17,425,170 | Mersenne | ^{[19]} |
5 | 2^{43112609} − 1 | 2008-08-23 | 12,978,189 | Mersenne | ^{[20]} |
6 | 2^{42643801} − 1 | 2009-06-04 | 12,837,064 | Mersenne | ^{[21]} |
7 | 2^{37156667} − 1 | 2008-09-06 | 11,185,272 | Mersenne | ^{[20]} |
8 | 2^{32582657} − 1 | 2006-09-04 | 9,808,358 | Mersenne | ^{[22]} |
9 | 10223 × 2^{31172165} + 1 | 2016-10-31 | 9,383,761 | Proth | ^{[23]} |
10 | 2^{30402457} − 1 | 2005-12-15 | 9,152,052 | Mersenne | ^{[24]} |
11 | 2^{25964951} − 1 | 2005-02-18 | 7,816,230 | Mersenne | ^{[25]} |
12 | 2^{24036583} − 1 | 2004-05-15 | 7,235,733 | Mersenne | ^{[26]} |
13 | 202705 × 2^{21320516} + 1 | 2021-12-01 | 6,418,121 | Proth | ^{[27]} |
14 | 2^{20996011} − 1 | 2003-11-17 | 6,320,430 | Mersenne | ^{[28]} |
15 | 1059094^{1048576} + 1 | 2018-10-31 | 6,317,602 | Generalized Fermat | ^{[29]} |
16 | 919444^{1048576} + 1 | 2017-08-29 | 6,253,210 | Generalized Fermat | ^{[30]} |
17 | 168451 × 2^{19375200} + 1 | 2017-09-17 | 5,832,522 | Proth | ^{[31]} |
18 | 69 × 2^{18831865} − 1 | 2021-12-16 | 5,668,959 | ||
19 | 7 × 2^{18233956} + 1 | 2020-10-01 | 5,488,969 | Proth | ^{[32]} |
20 | 3 × 2^{17748034} − 1 | 2021-09-06 | 5,342,692 | 321 | ^{[33]} |