The largest known top number (as of January 2022) is 2^{82,589,933} − 1, a number which has 24,862,048 digits when written in immoral 10. It used to be chanced on by technique of a pc volunteered by Patrick Laroche of the Wide Net Mersenne High Search (GIMPS) in 2018.^{[1]}
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add advertising hereA 2020 location of the alternative of digits in largest known top by one year, since the electronic pc. The vertical scale is logarithmic.
A top number is a obvious integer, excluding 1, without a divisors diversified than 1 and itself. Basically based on Euclid’s theorem there are infinitely many top numbers, so there might be not any such thing as a largest top.
Many of the largest known primes are Mersenne primes, numbers that are one no longer as much as a vitality of two, on fable of they’ll utilise a specialised primality take a look at that’s quicker than the classic one. As of December 2020, the eight largest known primes are Mersenne primes.^{[2]} The final seventeen memoir primes were Mersenne primes.^{[3]}^{[4]} The binary representation of any Mersenne top is mild of all 1’s, since the binary get of two^{k} − 1 is purely k 1’s.^{[5]}
The quick Fourier transform implementation of the Lucas–Lehmer primality take a look at for Mersenne numbers is terribly snappy when in comparison with diversified known primality tests for diversified forms of numbers. With most unique pc programs, a multimillion digit Mersennedelight in number would be confirmed top, however handiest multithousand digit diversified numbers would be confirmed top. Attainable primes, such because the immoral10 repunit R_{8177207}, cross probabilistic primality tests however are no longer indubitably confirmed top.
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add advertising hereMost unique memoir[edit]
The memoir is for the time being held by 2^{82,589,933} − 1 with 24,862,048 digits, chanced on by GIMPS in December 2018.^{[1]} The first and final 120 digits of its sign are confirmed below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 …
(24,861,808 digits pushed aside)
… 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591^{[6]}
Prizes[edit]
The Wide Net Mersenne High Search (GIMPS) for the time being offers a US$3,000 study discovery award for members who gather and trudge their free instrument and whose pc discovers a brand novel Mersenne top having fewer than 100 million digits.
There are several prizes equipped by the Electronic Frontier Foundation for memoir primes.^{[7]} GIMPS is additionally coordinating its prolongedfluctuate search efforts for primes of 100 million digits and bigger and could additionally fair split the Electronic Frontier Foundation’s US$150,000 prize with a winning participant.
The memoir passed one million digits in 1999, incomes a US$50,000 prize.^{[8]} In 2008, the memoir passed ten million digits, incomes a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.^{[7]} Time known as it the 29th top invention of 2008.^{[9]} Each the US$50,000 and the US$100,000 prizes were received by participation in GIMPS. Extra prizes are being equipped for the principle top number chanced on with on the least one hundred million digits and the principle with on the least one billion digits.^{[7]}
Historical previous of largest known top numbers[edit]
Commemorative postmark aged by the UIUC Math Department after proving that M_{11213} is top
The next table lists the progression of the largest known top number in ascending reveal.^{[3]} Here M_{p} = 2^{p} − 1 is the Mersenne number with exponent p. The longest memoirholder known used to be M_{19} = 524,287, which used to be the largest known top for 144 years. No records are known sooner than 1456.
Number  Decimal expansion (handiest for numbers < M_{1000}) 
Digits  one year chanced on  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 no longer explicitly post the primality of 6,700,417, however the ways he had aged to factorise 2^{32} + 1 intended that he had already accomplished numerous the work most essential to level this, and some experts middle of attention on he knew of it.^{[10]} 

M_{31}  2,147,483,647  10  1772  Leonhard Euler 
999,999,000,001  12  1851  Included (however demandmarked) in a checklist of primes by Looff. Given his uncertainty, some attain no longer consist of this as a memoir.  
67,280,421,310,721  14  1855  Thomas Clausen (however no proof used to be equipped).  
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 memoir no longer build by pc.  
180×(M_{127})^{2}+1 
521064401567922879406069432539 
79  1951  J. C. P. Miller & D. J. Wheeler^{[11]} Utilizing Cambridge’s EDSAC pc 
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 crew, “Amdahl Six”: John Brown, Landon Curt Noll, B. Okay. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.^{[13]}^{[14]} Ideal nonMersenne top that used to be the largest known top when it used to be chanced on. 

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, College 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 Trek 
M_{82589933}  148894445742…325217902591  24,862,048  2018  GIMPS, Patrick Laroche 
GIMPS chanced on the fifteen most unique records (all of them Mersenne primes) on extraordinary pc programs operated by members at some level of the field.
The twenty largest known top numbers[edit]
A checklist of the 5,000 largest known primes is maintained by Chris Okay. Caldwell,^{[15]}^{[16]} of which the twenty largest are listed below.
Obnoxious  Number  Found out  Digits  Comprise  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  202705 × 2^{21320516} + 1  20211201  6,418,121  Proth  ^{[27]} 
14  2^{20996011} − 1  20031117  6,320,430  Mersenne  ^{[28]} 
15  1059094^{1048576} + 1  20181031  6,317,602  Generalized Fermat  ^{[29]} 
16  919444^{1048576} + 1  20170829  6,253,210  Generalized Fermat  ^{[30]} 
17  168451 × 2^{19375200} + 1  20170917  5,832,522  Proth  ^{[31]} 
18  69 × 2^{18831865} − 1  20211216  5,668,959  
19  7 × 2^{18233956} + 1  20201001  5,488,969  Proth  ^{[32]} 
20  3 × 2^{18196595} − 1  20220118  5,477,722  321 
Notice additionally[edit]
References[edit]
 ^ ^{a} ^{b} ^{c} “GIMPS Project Discovers Ideal Recognized High Number: 2^{82,589,933}1″. Mersenne Be taught, Inc. 21 December 2018. Retrieved 21 December 2018.
 ^ Caldwell, Chris. “The ideally superior known primes – Database Search Output”. High Pages. Retrieved June 3, 2018.
 ^ ^{a} ^{b} Caldwell, Chris. “The Ideal Recognized High by one year: A Short Historical previous”. High Pages. Retrieved January 20, 2016.
 ^ The final nonMersenne to be the largest known top, used to be 391,581 ⋅ 2^{216,193} − 1; investigate crosstake a look at additionally The Ideal Recognized High by one year: A Short Historical previous by Caldwell.
 ^ “Ideal Numbers”. Penn Divulge College. Retrieved 6 October 2019.
A sharp facet show is ready the binary representations of those numbers…
 ^ “51st Recognized Mersenne High Found out”.
 ^ ^{a} ^{b} ^{c} “File 12MillionDigit High Number Nets $100,000 Prize”. Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
 ^ Electronic Frontier Foundation, Wide High Nets Wide Prize.
 ^ “Ideal Inventions of 2008 – 29. The 46th Mersenne High”. Time. Time Inc. October 29, 2008. Archived from the distinctive on November 2, 2008. Retrieved January 17, 2012.
 ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even Extra. ISBN 9780883855843.
 ^ J. Miller, Tremendous High Numbers. Nature 168, 838 (1951).
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Landon Curt Noll, Tremendous High Number Found out by SGI/Cray Supercomputer.
 ^ Letters to the Editor. The American Mathematical Monthtomonth 97, no. 3 (1990), p. 214. Accessed Would possibly more than doubtless additionally 22, 2020.
 ^ Proofcode: Z, The High Pages.
 ^ “The High Database: The List of Ideal Recognized Primes Home Net page”. primes.utm.edu/primes. Chris Okay. Caldwell. Retrieved 30 September 2017.
 ^ “The High Twenty: Ideal Recognized Primes”. Chris Okay. Caldwell. Retrieved 3 January 2018.
 ^ “GIMPS Project Discovers Ideal Recognized High Number: 2^{77,232,917}1″. mersenne.org. Wide Net Mersenne High Search. Retrieved 3 January 2018.
 ^ “GIMPS Project Discovers Ideal Recognized High Number: 2^{74,207,281}1″. mersenne.org. Wide Net Mersenne High Search. Retrieved 29 September 2017.
 ^ “GIMPS Discovers 48th Mersenne High, 2^{57,885,161}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High 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 Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 15 September 2008. Retrieved 29 September 2017.
 ^ “GIMPS Discovers 47th Mersenne High, 2^{42,643,801}1 is latest, however no longer the largest, known Mersenne High”. mersenne.org. Wide Net Mersenne High Search. 12 April 2009. Retrieved 29 September 2017.
 ^ “GIMPS Discovers 44th Mersenne High, 2^{32,582,657}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High 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 High, 2^{30,402,457}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 24 December 2005. Retrieved 29 September 2017.
 ^ “GIMPS Discovers 42nd Mersenne High, 2^{25,964,951}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 27 February 2005. Retrieved 29 September 2017.
 ^ “GIMPS Discovers 41st Mersenne High, 2^{24,036,583}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 28 Would possibly more than doubtless additionally 2004. Retrieved 29 September 2017.
 ^ “PrimeGrid’s Prolonged Sierpinski Be troubled High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 28 December 2021.
 ^ “GIMPS Discovers 40th Mersenne High, 2^{20,996,011}1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 2 December 2003. Retrieved 29 September 2017.
 ^ “PrimeGrid’s Generalized Fermat High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
 ^ “PrimeGrid’s Generalized Fermat High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
 ^ “PrimeGrid’s High Sierpinski Be troubled” (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.
 ^ “PrimePage Primes: 7 x 2^18233956 + 1”. Retrieved 10 February 2021.
External hyperlinks[edit]
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