Largest Known Prime Number
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Largest Known Prime Number

The largest known prime number (as of December 2020) is , 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 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, excluding 1, with no divisors other than 1 and itself. 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 2020, 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 2k - 1 is simply k 1's.[5]

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 with 24,862,048 digits, found by GIMPS in December 2018.[1] Its value is:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...

(24,861,808 digits omitted)

... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]

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.[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]

## History of largest known prime numbers

The following table lists the progression of the largest known prime number in ascending order.[3] Here is the Mersenne number with exponent n. The longest record-holder known was , 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
M13 8,191 4 1456 Anonymous
M17 131,071 6 1588 Pietro Cataldi
M19 524,287 6 1588 Pietro Cataldi
${\displaystyle {\tfrac {2^{32}+1}{641}}}$ 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.[10]
M31 2,147,483,647 10 1772 Leonhard Euler
${\displaystyle {\tfrac {2^{64}+1}{274177}}}$ 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
${\displaystyle {\tfrac {2^{148}+1}{17}}}$ 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[11]
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[12]
M23209 6,987 1979 Landon Curt Noll[12]
M44497 13,395 1979 David Slowinski and Harry L. Nelson[12]
M86243 25,962 1982 David Slowinski[12]
M132049 39,751 1983 David Slowinski[12]
M216091 65,050 1985 David Slowinski[12]
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.[13][14]
Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839 227,832 1992 David Slowinski and Paul Gage[12]
M859433 258,716 1994 David Slowinski and Paul Gage[12]
M1257787 378,632 1996 David Slowinski and Paul Gage[12]
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,[15][16] of which the twenty largest are listed below.

Rank Number Discovered Digits Ref
1 282589933 - 1 2018-12-07 24,862,048 [1]
2 277232917 - 1 2017-12-26 23,249,425 [17]
3 274207281 - 1 2016-01-07 22,338,618 [18]
4 257885161 - 1 2013-01-25 17,425,170 [19]
5 243112609 - 1 2008-08-23 12,978,189 [20]
6 242643801 - 1 2009-06-04 12,837,064 [21]
7 237156667 - 1 2008-09-06 11,185,272 [20]
8 232582657 - 1 2006-09-04 9,808,358 [22]
9 10223 × 231172165 + 1 2016-10-31 9,383,761 [23]
10 230402457 - 1 2005-12-15 9,152,052 [24]
11 225964951 - 1 2005-02-18 7,816,230 [25]
12 224036583 - 1 2004-05-15 7,235,733 [26]
13 220996011 - 1 2003-11-17 6,320,430 [27]
14 10590941048576 + 1 2018-10-31 6,317,602 [28]
15 9194441048576 + 1 2017-08-29 6,253,210 [29]
16 168451 × 219375200 + 1 2017-09-17 5,832,522 [30]
17 1234471048576 - 123447524288 + 1 2017-02-23 5,338,805 [31]
18 7 × 66772401 + 1 2019-09-09 5,269,954 [32]
19 8508301 × 217016603 - 1 2018-03-21 5,122,515 [33]
20 6962 × 312863120 - 1 2020-02-29 4,269,952 [34]

## References

1. ^ a b c "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 2018.
2. ^ Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved 2018.
3. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 2016.
4. ^ The last non-Mersenne to be the largest known prime, was 391,581 ? 2216,193 - 1; see also The Largest Known Prime by Year: A Brief History by Caldwell.
5. ^ "Perfect Numbers". Penn State University. Retrieved 2019. An interesting side note is about the binary representations of those numbers...
6. ^ https://www.mersenne.org/primes/press/M82589933.html
7. ^ a b c "Record 12-Million-Digit Prime Number Nets \$100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved 2011.
8. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
9. ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Retrieved 2012.
10. ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. ISBN 9780883855843.
11. ^ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
12. ^ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
13. ^ Proof-code: Z, The Prime Pages.
14. ^ "The Prime Database: The List of Largest Known Primes Home Page". primes.utm.edu/primes. Chris K. Caldwell. Retrieved 2017.
15. ^ "The Top Twenty: Largest Known Primes". Chris K. Caldwell. Retrieved 2018.
16. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 2018.
17. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 2017.
18. ^ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 2017.
19. ^ a b "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 2017.
20. ^ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 2017.
21. ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 2017.
22. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 2017.
23. ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 2017.
24. ^ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 2017.
25. ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 2017.
26. ^ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 2017.
27. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 2018.
28. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 2017.
29. ^ "PrimeGrid's Prime Sierpinski Problem" (PDF). primegrid.com. PrimeGrid. Retrieved 2017.
30. ^ "The Prime Database: Phi(3,-123447^524288)". primes.utm.edu. The Prime Pages. Retrieved 2017.
31. ^ "The Prime Database: 7*6^6772401+1". primes.utm.edu. The Prime Pages=12 September 2019.
32. ^ "PrimeGrid's Woodall Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 2018.
33. ^ "The Prime Database: 6962*31^2863120-1". primes.utm.edu. The Prime Pages. Retrieved 2020.