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# Ideal Recognized High Number

76

The largest known top number (as of January 2022) is 282,589,933 − 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]

A 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 twok − 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 multi-million digit Mersenne-delight in number would be confirmed top, however handiest multi-thousand digit diversified numbers would be confirmed top. Attainable primes, such because the immoral-10 repunit R8177207, cross probabilistic primality tests however are no longer indubitably confirmed top.

## Most unique memoir

The memoir is for the time being held by 282,589,933 − 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

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 prolonged-fluctuate 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

Commemorative postmark aged by the UIUC Math Department after proving that M11213 is top

The next table lists the progression of the largest known top number in ascending reveal.[3] Here Mp = 2p − 1 is the Mersenne number with exponent p. The longest memoir-holder known used to be M19 = 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 < M1000)
Digits one year chanced on Discoverer
M13 8,191 4 1456 Anonymous
M17 131,071 6 1588 Pietro Cataldi
M19 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 232 + 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]
M31 2,147,483,647 10 1772 Leonhard Euler
999,999,000,001 12 1851 Included (however demand-marked) 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).
M127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 Édouard Lucas
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 memoir no longer build by pc.
180×(M127)2+1

5210644015679228794060694325390955853335898483908056458352183851018372555735221

79 1951 J. C. P. Miller & D. J. Wheeler[11]
Utilizing Cambridge’s EDSAC pc
M521

6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151

157 1952
M607

531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127

183 1952
M1279 104079321946…703168729087 386 1952
M2203 147597991521…686697771007 664 1952
M2281 446087557183…418132836351 687 1952
M3217 259117086013…362909315071 969 1957
M4423 285542542228…902608580607 1,332 1961
M9689 478220278805…826225754111 2,917 1963
M9941 346088282490…883789463551 2,993 1963
M11213 281411201369…087696392191 3,376 1963
M19937 431542479738…030968041471 6,002 1971 Bryant Tuckerman
M21701 448679166119…353511882751 6,533 1978 Laura A. Nickel and Landon Curt Noll[12]
M23209 402874115778…523779264511 6,987 1979 Landon Curt Noll[12]
M44497 854509824303…961011228671 13,395 1979 David Slowinski and Harry L. Nelson[12]
M86243 536927995502…709433438207 25,962 1982 David Slowinski[12]
M132049 512740276269…455730061311 39,751 1983 David Slowinski[12]
M216091 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 non-Mersenne top that used to be the largest known top when it used to be chanced on.
M756839 174135906820…328544677887 227,832 1992 David Slowinski and Paul Gage[12]
M859433 129498125604…243500142591 258,716 1994 David Slowinski and Paul Gage[12]
M1257787 412245773621…976089366527 378,632 1996 David Slowinski and Paul Gage[12]
M1398269 814717564412…868451315711 420,921 1996 GIMPS, Joel Armengaud
M2976221 623340076248…743729201151 895,932 1997 GIMPS, Gordon Spence
M3021377 127411683030…973024694271 909,526 1998 GIMPS, Roland Clarkson
M6972593 437075744127…142924193791 2,098,960 1999 GIMPS, Nayan Hajratwala
M13466917 924947738006…470256259071 4,053,946 2001 GIMPS, Michael Cameron
M20996011 125976895450…762855682047 6,320,430 2003 GIMPS, Michael Shafer
M24036583 299410429404…882733969407 7,235,733 2004 GIMPS, Josh Findley
M25964951 122164630061…280577077247 7,816,230 2005 GIMPS, Martin Nowak
M30402457 315416475618…411652943871 9,152,052 2005 GIMPS, College of Central Missouri professors Curtis Cooper and Steven Boone
M32582657 124575026015…154053967871 9,808,358 2006 GIMPS, Curtis Cooper and Steven Boone
M43112609 316470269330…166697152511 12,978,189 2008 GIMPS, Edson Smith
M57885161 581887266232…071724285951 17,425,170 2013 GIMPS, Curtis Cooper
M74207281 300376418084…391086436351 22,338,618 2016 GIMPS, Curtis Cooper
M77232917 467333183359…069762179071 23,249,425 2017 GIMPS, Jonathan Trek
M82589933 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

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 282589933 − 1 2018-12-07 24,862,048 Mersenne [1]
2 277232917 − 1 2017-12-26 23,249,425 Mersenne [17]
3 274207281 − 1 2016-01-07 22,338,618 Mersenne [18]
4 257885161 − 1 2013-01-25 17,425,170 Mersenne [19]
5 243112609 − 1 2008-08-23 12,978,189 Mersenne [20]
6 242643801 − 1 2009-06-04 12,837,064 Mersenne [21]
7 237156667 − 1 2008-09-06 11,185,272 Mersenne [20]
8 232582657 − 1 2006-09-04 9,808,358 Mersenne [22]
9 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [23]
10 230402457 − 1 2005-12-15 9,152,052 Mersenne [24]
11 225964951 − 1 2005-02-18 7,816,230 Mersenne [25]
12 224036583 − 1 2004-05-15 7,235,733 Mersenne [26]
13 202705 × 221320516 + 1 2021-12-01 6,418,121 Proth [27]
14 220996011 − 1 2003-11-17 6,320,430 Mersenne [28]
15 10590941048576 + 1 2018-10-31 6,317,602 Generalized Fermat [29]
16 9194441048576 + 1 2017-08-29 6,253,210 Generalized Fermat [30]
17 168451 × 219375200 + 1 2017-09-17 5,832,522 Proth [31]
18 69 × 218831865 − 1 2021-12-16 5,668,959
19 7 × 218233956 + 1 2020-10-01 5,488,969 Proth [32]
20 3  ×  218196595 − 1 2022-01-18 5,477,722 321

## References

1. ^ a b c “GIMPS Project Discovers Ideal Recognized High Number: 282,589,933-1″. Mersenne Be taught, Inc. 21 December 2018. Retrieved 21 December 2018.
2. ^ Caldwell, Chris. “The ideally superior known primes – Database Search Output”. High Pages. Retrieved June 3, 2018.
3. ^ a b Caldwell, Chris. “The Ideal Recognized High by one year: A Short Historical previous”. High Pages. Retrieved January 20, 2016.
4. ^ The final non-Mersenne to be the largest known top, used to be 391,581 ⋅ 2216,193 − 1; investigate cross-take a look at additionally The Ideal Recognized High by one year: A Short Historical previous by Caldwell.
5. ^ “Ideal Numbers”. Penn Divulge College. Retrieved 6 October 2019. A sharp facet show is ready the binary representations of those numbers…
6. ^ “51st Recognized Mersenne High Found out”.
7. ^ a b c “File 12-Million-Digit High Number Nets \$100,000 Prize”. Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
8. ^ Electronic Frontier Foundation, Wide High Nets Wide Prize.
9. ^ “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.
10. ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even Extra. ISBN 9780883855843.
11. ^ J. Miller, Tremendous High Numbers. Nature 168, 838 (1951).
12. ^ a b c d e f g h i Landon Curt Noll, Tremendous High Number Found out by SGI/Cray Supercomputer.
13. ^ Letters to the Editor. The American Mathematical Month-to-month 97, no. 3 (1990), p. 214. Accessed Would possibly more than doubtless additionally 22, 2020.
14. ^ Proof-code: Z, The High Pages.
15. ^ “The High Database: The List of Ideal Recognized Primes Home Net page”. primes.utm.edu/primes. Chris Okay. Caldwell. Retrieved 30 September 2017.
16. ^ “The High Twenty: Ideal Recognized Primes”. Chris Okay. Caldwell. Retrieved 3 January 2018.
17. ^ “GIMPS Project Discovers Ideal Recognized High Number: 277,232,917-1″. mersenne.org. Wide Net Mersenne High Search. Retrieved 3 January 2018.
18. ^ “GIMPS Project Discovers Ideal Recognized High Number: 274,207,281-1″. mersenne.org. Wide Net Mersenne High Search. Retrieved 29 September 2017.
19. ^ “GIMPS Discovers 48th Mersenne High, 257,885,161-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 5 February 2013. Retrieved 29 September 2017.
20. ^ a b “GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 15 September 2008. Retrieved 29 September 2017.
21. ^ “GIMPS Discovers 47th Mersenne High, 242,643,801-1 is latest, however no longer the largest, known Mersenne High”. mersenne.org. Wide Net Mersenne High Search. 12 April 2009. Retrieved 29 September 2017.
22. ^ “GIMPS Discovers 44th Mersenne High, 232,582,657-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 11 September 2006. Retrieved 29 September 2017.
23. ^ “PrimeGrid’s Seventeen or Bust Subproject” (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
24. ^ “GIMPS Discovers 43rd Mersenne High, 230,402,457-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 24 December 2005. Retrieved 29 September 2017.
25. ^ “GIMPS Discovers 42nd Mersenne High, 225,964,951-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 27 February 2005. Retrieved 29 September 2017.
26. ^ “GIMPS Discovers 41st Mersenne High, 224,036,583-1 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.
27. ^ “PrimeGrid’s Prolonged Sierpinski Be troubled High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 28 December 2021.
28. ^ “GIMPS Discovers 40th Mersenne High, 220,996,011-1 is now the Ideal Recognized High”. mersenne.org. Wide Net Mersenne High Search. 2 December 2003. Retrieved 29 September 2017.
29. ^ “PrimeGrid’s Generalized Fermat High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
30. ^ “PrimeGrid’s Generalized Fermat High Search” (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
31. ^ “PrimeGrid’s High Sierpinski Be troubled” (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.
32. ^ “PrimePage Primes: 7 x 2^18233956 + 1”. Retrieved 10 February 2021.

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