Largest known prime number

The largest known prime number (as of October 2020) 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.

A 2020 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic.

A prime number is a positive integer with no divisors other than 1 and itself, excluding 1. Euclid recorded a proof that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.

Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two. As of December 2018, the eight largest known primes are Mersenne primes. The last seventeen record primes were Mersenne primes. 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.

The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers.

Current record

The record is currently held by 2^82,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018. Its value is:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...

(24,861,808 digits omitted)

... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591

The first and last 120 digits are shown above.

Prizes

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. 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. In 2008, the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation. Time called it the 29th top invention of 2008. 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.

History of largest known prime numbers

The following table lists the progression of the largest known prime number in ascending order. Here M_n = 2ⁿ − 1 is the Mersenne number with exponent n. The longest record-holder known was M₁₉ = 524,287, which was the largest known prime for 144 years. No records are known before 1456.

Number	Decimal expansion (only for numbers < M5000)	Digits	Year found	Discoverer (see also Mersenne prime)
M13	8,191	4	1456	Anonymous
M17	131,071	6	1588	Pietro Cataldi
M19	524,287	6	1588	Pietro Cataldi
${\tfrac {2^{32}+1}{641}}$	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 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.
M31	2,147,483,647	10	1772	Leonhard Euler
${\tfrac {2^{64}+1}{274177}}$	67,280,421,310,721	14	1855	Thomas Clausen
M127	170,141,183,460,469,231,731,687,303,715,884,105,727	39	1876	Édouard Lucas
${\tfrac {2^{148}+1}{17}}$	20,988,936,657,440,586,486,151,264,256,610,222,593,863,921	44	1951	Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1	5210644015679228794060694325390955853335898483908056458352 183851018372555735221	79	1951	J. C. P. Miller & D. J. Wheeler Using Cambridge's EDSAC computer
M521	6864797660130609714981900799081393217269435300143305409394 4634591855431833976560521225596406614545549772963113914808 58037121987999716643812574028291115057151	157	1952
M607	53113799281676709868958820655246862732959311772703192319944 4138200403559860852242739162502265229285668889329486246501 01534657933765270723940951997876658735194383127083539321903 1728127	183	1952
M1279	10407932194664399081925240327364085538615262247266704805319 112350403608059673360298012239441732324184842421613954281007 79138356624832346490813990660567732076292412950938922034577 318334966158355047295942054768981121169367714754847886696250 138443826029173234888531116082853841658502825560466622483189 091880184706822220314052102669843548873295802887805086973618 6900714720710555703168729087	386	1952
M2203	14759799152141802350848986227373817363120661453331697751477712 164785702978780789493774073370493892893827485075314964804772 8126483876025919181446336533026954049696120111343015690239609 398909022625932693502528140961498349938822283144859860183431 853623092377264139020949023183644689960821079548296376309423 6630945410832793769905399982457186322944729636418890623372171 723742105636440368218459649632948538696905872650486914434637 4575072804418236768135178520993486608471725794084223166780976 7022401199028017047489448742692474210882353680848507250224051 9452587542875349976558572670229633962575212637477897785501552 646522609988869914013540483809865681250419497686697771007	664	1952
M2281	446087557183758429571151706402101809886208632412859901111991219963404685792 82047336911254526900398902615324593112431670239575870569367936479090349746 114707106525419335393812497822630794731241079887486904007027932842881031175 484410809487825249486676096958699812898264587759602897917153696250306842 961733170218475032458300917183210491605015762888660637214550170222592512522 40768296054271735739648129952505694124807207384768552936816667128448311908 776206067866638621902401185707368319018864792258104147140789353865624979681 787291276295949244119609613867139462798992750069549171397587960612238033935 373810346664944029510520590479686932553886479304409251041868170096401717641 33172418132836351	687	1952
M3217	25911708601320262777624676792244153094181888755312542730397492316187401926658 63620862012095168004834065506952417331941774416895092388070174103777095975120 423130666240829163535179523111861548622656045476911275958487756105687579311910 17711408826252153849035830401185072116424747461823031471398340229288074545677 907941037288235820705892351068433882986888616658650280927692080339605869308 79050040950370987590211901837199162099400256893511313654882973911265679730324 19865172501164127035097054277734779723498216764434466683831193225400996489940 5179024162405651905448369080961606162574304236172186333941585242643120873726 6591962061753535748892894599629195183082621860853400937932839420261866586142 50325145077309627423537682293864940712770084607712421182308080413929808705750 47138252645714483793711250320818261265666490842516994539518877896136502484057 3937859459944433523118828012366040626246860921215034993758478229223714433962 8858485938215738821232393687046160677362909315071	969	1957
M4423	2855425422282796139015635661021640083261642386447028891992474566022844003906 00653875954571505539843239754513915896150297878399377056071435169747221107988 7911982009884775313392142827720160590099045866862549890848157354224804090223 44297588352526004383890632616124076317387416881148592486188361873904175783145 6960169195743907655982801885990355784485910776836771755204340742877265780062 66759615970759521327828555662781678385691581844436444812511562428136742490459 363212810180276096088111401003377570363545725120924073646921576797146199387619 29656030268026179011813292501232304644443862230887792460937377301248168167242 44936744744885377701557830068808526481615130671448147902883666640622572746652 757871273746492310963750011709018907862633246195787957314256938050730561196775 8033808433338198750090296883193591309526982131114132239335649017848872898228 81562826008138312961436638459454311440437538215428712777456064478585641592133 2844358020642271469491309176271644704168967807009677359042980890961675045292 725800084350034483162829708990272864998199438764723457427626372969484830475 09171741861811306885187927486226122933413689280566343844666463265724761672756 60839105650528975713899320211121495795311427946254553305387067821067601768750 97786610046001460213840844802122505368905479374200309572209673295475072171811 5531871310231057902608580607	1,332	1961
M9689		2,917	1963
M9941		2,993	1963
M11213		3,376	1963
M19937		6,002	1971	Bryant Tuckerman
M21701		6,533	1978	Laura A. Nickel and Landon Curt Noll
M23209		6,987	1979	Landon Curt Noll
M44497		13,395	1979	David Slowinski and Harry L. Nelson
M86243		25,962	1982	David Slowinski
M132049		39,751	1983	David Slowinski
M216091		65,050	1985	David Slowinski
391581×2216193−1		65,087	1989	A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello. Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839		227,832	1992	David Slowinski and Paul Gage
M859433		258,716	1994	David Slowinski and Paul Gage
M1257787		378,632	1996	David Slowinski and Paul Gage
M1398269		420,921	1996	GIMPS, Joel Armengaud
M2976221		895,932	1997	GIMPS, Gordon Spence
M3021377		909,526	1998	GIMPS, Roland Clarkson
M6972593		2,098,960	1999	GIMPS, Nayan Hajratwala
M13466917		4,053,946	2001	GIMPS, Michael Cameron
M20996011		6,320,430	2003	GIMPS, Michael Shafer
M24036583		7,235,733	2004	GIMPS, Josh Findley
M25964951		7,816,230	2005	GIMPS, Martin Nowak
M30402457		9,152,052	2005	GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M32582657		9,808,358	2006	GIMPS, Curtis Cooper and Steven Boone
M43112609		12,978,189	2008	GIMPS, Edson Smith
M57885161		17,425,170	2013	GIMPS, Curtis Cooper
M74207281		22,338,618	2016	GIMPS, Curtis Cooper
M77232917		23,249,425	2017	GIMPS, Jonathan Pace
M82589933		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.

The twenty largest known prime numbers

A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, of which the twenty largest are listed below.

Rank	Number	Discovered	Digits	Ref
1	282589933 − 1	2018-12-07	24,862,048
2	277232917 − 1	2017-12-26	23,249,425
3	274207281 − 1	2016-01-07	22,338,618
4	257885161 − 1	2013-01-25	17,425,170
5	243112609 − 1	2008-08-23	12,978,189
6	242643801 − 1	2009-06-04	12,837,064
7	237156667 − 1	2008-09-06	11,185,272
8	232582657 − 1	2006-09-04	9,808,358
9	10223 × 231172165 + 1	2016-10-31	9,383,761
10	230402457 − 1	2005-12-15	9,152,052
11	225964951 − 1	2005-02-18	7,816,230
12	224036583 − 1	2004-05-15	7,235,733
13	220996011 − 1	2003-11-17	6,320,430
14	10590941048576 + 1	2018-10-31	6,317,602
15	9194441048576 + 1	2017-08-29	6,253,210
16	168451 × 219375200 + 1	2017-09-17	5,832,522
17	1234471048576 − 123447524288 + 1	2017-02-23	5,338,805
18	7 × 66772401 + 1	2019-09-09	5,269,954
19	8508301 × 217016603 − 1	2018-03-21	5,122,515
20	6962 × 312863120 − 1	2020-02-29	4,269,952

See also

• Mersenne prime
• Primality test
• Prime number
• Generalized Fermat prime
• Cullen number
• Woodall number
• Titanic prime
• Gigantic prime
• Megaprime
• Sophie Germain prime

External links

• Press release about the largest known prime 2^82,589,933−1
• Press release about the former largest known prime 2^77,232,917−1
• Press release about the former largest known prime 2^74,207,281−1