Largest known prime number
The largest known prime number (as of September 2021^{ [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. 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 multimillion digit Mersennelike number can be proven prime, but only multithousand digit other numbers can be proven prime. Probable primes, such as repunit R_{8177207}, pass probabilistic primality tests but are not truly proven prime.
Current record
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]}
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 longrange 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 M_{p} = 2^{p} − 1 is the Mersenne number with exponent p. The longest recordholder 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 (see also Mersenne prime) 

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 questionmarked) 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 nonMersenne 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.
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  Form  Ref 

1  2^{82589933} − 1  20181207  24,862,048  Mersenne  ^{ [1]} 
2  2^{77232917} − 1  20171226  23,249,425  Mersenne  ^{ [17]} 
3  2^{74207281} − 1  20160107  22,338,618  Mersenne  ^{ [18]} 
4  2^{57885161} − 1  20130125  17,425,170  Mersenne  ^{ [19]} 
5  2^{ 43112609} − 1  20080823  12,978,189  Mersenne  ^{ [20]} 
6  2^{42643801} − 1  20090604  12,837,064  Mersenne  ^{ [21]} 
7  2^{37156667} − 1  20080906  11,185,272  Mersenne  ^{ [20]} 
8  2^{32582657} − 1  20060904  9,808,358  Mersenne  ^{ [22]} 
9  10223 × 2^{31172165} + 1  20161031  9,383,761  Proth  ^{ [23]} 
10  2^{30402457} − 1  20051215  9,152,052  Mersenne  ^{ [24]} 
11  2^{25964951} − 1  20050218  7,816,230  Mersenne  ^{ [25]} 
12  2^{24036583} − 1  20040515  7,235,733  Mersenne  ^{ [26]} 
13  2^{20996011} − 1  20031117  6,320,430  Mersenne  ^{ [27]} 
14  1059094^{1048576} + 1  20181031  6,317,602  Generalized Fermat  ^{ [28]} 
15  919444^{1048576} + 1  20170829  6,253,210  Generalized Fermat  ^{ [29]} 
16  168451 × 2^{19375200} + 1  20170917  5,832,522  Proth  ^{ [30]} 
17  7 × 2^{18233956} + 1  20201001  5,488,969  Proth  ^{ [31]} 
18  3 × 2^{17748034} − 1  20210906  5,342,692  321  ^{ [32]} 
19  123447^{1048576} − 123447^{524288} + 1  20170223  5,338,805  Generalized unique  ^{ [33]} 
20  7 × 6^{6772401} + 1  20190909  5,269,954  Generalized Proth  ^{ [34]} 
See also
References
 ^ ^{a} ^{b} ^{c} "GIMPS Project Discovers Largest Known Prime Number: 2^{82,589,933}1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
 ^ Caldwell, Chris. "The largest known primes  Database Search Output". Prime Pages. Retrieved June 3, 2018.
 ^ ^{a} ^{b} Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016.
 ^ The last nonMersenne to be the largest known prime, was 391,581 ⋅ 2^{216,193} − 1; see also The Largest Known Prime by year: A Brief History by Caldwell.

^
"Perfect Numbers". Penn State University. Retrieved 6 October 2019.
An interesting side note is about the binary representations of those numbers...
 ^ "51st Known Mersenne Prime Discovered".
 ^ ^{a} ^{b} ^{c} "Record 12MillionDigit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
 ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
 ^ "Best Inventions of 2008  29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
 ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. ISBN 9780883855843.
 ^ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
 ^ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
 ^ Proofcode: Z, The Prime Pages.
 ^ "The Prime Database: The List of Largest Known Primes Home Page". primes.utm.edu/primes. Chris K. Caldwell. Retrieved 30 September 2017.
 ^ "The Top Twenty: Largest Known Primes". Chris K. Caldwell. Retrieved 3 January 2018.
 ^ "GIMPS Project Discovers Largest Known Prime Number: 2^{77,232,917}1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
 ^ "GIMPS Project Discovers Largest Known Prime Number: 2^{74,207,281}1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 48th Mersenne Prime, 2^{57,885,161}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
 ^ ^{a} ^{b} "GIMPS Discovers 45th and 46th Mersenne Primes, 2^{43,112,609}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 47th Mersenne Prime, 2^{42,643,801}1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 44th Mersenne Prime, 2^{32,582,657}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
 ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
 ^ "GIMPS Discovers 43rd Mersenne Prime, 2^{30,402,457}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 42nd Mersenne Prime, 2^{25,964,951}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 41st Mersenne Prime, 2^{24,036,583}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
 ^ "GIMPS Discovers 40th Mersenne Prime, 2^{20,996,011}1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.
 ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
 ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
 ^ "PrimeGrid's Prime Sierpinski Problem" (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.
 ^ "PrimePage Primes: 7 x 2^18233956 + 1". Retrieved 10 February 2021.
 ^ "The Prime Database: 3*2^177480341". primes.utm.edu. The Prime Pages. Retrieved 13 September 2021.
 ^ "The Prime Database: Phi(3,123447^524288)". primes.utm.edu. The Prime Pages. Retrieved 30 September 2017.
 ^ "The Prime Database: 7*6^6772401+1". primes.utm.edu. The Prime Pages=12 September 2019.