已知最大素数

已知最大素数(截至2021年9月 (2021-09))为282,589,933 − 1,十进制时有24,862,048位数,由互联网梅森素数大搜索GIMPS)的志愿者派翠克·拉罗次(Patrick Laroche)于2018年发现[1]

电子电脑发明后已知最大素数数位的按年图表,至2016年为止。垂直刻度为对数尺度,红线为曲线拟合y = exp(0.187394 t − 360.527),t为年份

素数,又名素数,是一个除1与自身之外没有其他约数正整数欧几里得定理说明素数没有上限,不少数学家与嗜好者故一直寻找大素数。

不少大素数为梅森素数,定义为2的幂减去1的正整数。截至2018年12月 (2018-12),首八个已知大素数皆为梅森素数[2]。近十七次最大素数纪录皆为梅森素数[3][4]。所有梅森素数的二进制表示中,所有数字皆为1[5]

卢卡斯-莱默检验法快速傅里叶变换比起其他方式能更快速寻找到梅森素数

现时纪录

已知最大素数为282,589,933 − 1,共有24,862,048位数,由互联网梅森素数大搜索于2018年12月发现[1]。其数值为:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...

(省略24,861,808位数)

... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]

上面只显示首尾各120位数。

奖金

互联网梅森素数大搜索现为下载其软件并成功寻找新梅森素数的参与者提供3,000美元奖金,该梅森素数的数位应少于一亿位。

电子前哨基金会亦为大素数的找寻设立了数个奖项[7],互联网梅森素数大搜索亦有协调一亿数位以上的素数搜索,并与成功寻找者分享电子前哨基金会所提供的150,000元美金奖金。

1999年发现首个超过一百万数位的素数,并取得50,000美元奖金[8]。2008年发现了超过一千万数位的素数,并取得100,000美元奖金[7]时代杂志称之为2008年第29名最佳发现[9]两项奖金皆为互联网梅森素数大搜索的参加者。电子前哨基金会现为首个一亿及十亿数位的素数提供奖金[7]

已知最大素数历史

下表列出已知最大素数沿革,并按时序排列[3]。此处Mn = 2n − 1,为2的n次方。时间最长的纪录保持者为M19 = 524,287,为已知最大素数共计144年。1456年之前未存有关最大素数的纪录。

数字 数字展开
(仅限小于M5000的数字)
数位 发现年份 发现者
M13 8,191 4 1456 佚名
M17 131,071 6 1588 伯多禄·卡塔迪
M19 524,287 6 1588 伯多禄·卡塔迪
  6,700,417 7 1732 莱昂哈德·欧拉
欧拉并未正式发表此数,但他于232 + 1的因式分解中已完成此素数的大部分证明过程,故部分专家认为欧拉知道此为素数[10]
M31 2147483647 10 1772 莱昂哈德·欧拉
  67,280,421,310,721 14 1855 汤马斯·克劳森英语Thomas Clausen (mathematician)
M127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 爱德华·卢卡斯
  20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 44 1951 Aimé Ferrier
使用机械计算机发现,非使用电脑发现的最大素数
180×(M127)2+1 5210644015679228794060694325390955853335898483908056458352

183851018372555735221

79 1951 J. C. P.米勒英语J. C. P. Miller大卫·惠勒英语David Wheeler (computer scientist)[11]
使用剑桥大学英语University of Cambridge Mathematical LaboratoryEDSAC电脑
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
M21701 6,533 1978
M23209 6,987 1979
M44497 13,395 1979
M86243 25,962 1982
M132049 39,751 1983
M216091 65,050 1985
391581×2216193−1 65,087 1989 群组发现,包括约翰·布朗、蓝登·克特·诺尔英语Landon Curt Noll、B. K. 柏拉狄、哲恩·史密夫、乔尔·史密夫、沙治奥[12][13],为已知最大素数历史中最大的非梅森素数。
M756839 227,832 1992
M859433 258,716 1994
M1257787 378,632 1996
M1398269 420,921 1996 互联网梅森素数大搜索,乔尔·阿孟较得
M2976221 895,932 1997 互联网梅森素数大搜索,戈登·斯彭斯
M3021377 909,526 1998 互联网梅森素数大搜索,罗兰·克拉克森
M6972593 2,098,960 1999 互联网梅森素数大搜索,拿恩·哈拉华拉
M13466917 4,053,946 2001 互联网梅森素数大搜索,米高·卡梅伦
M20996011 6,320,430 2003 互联网梅森素数大搜索,米高·沙夫
M24036583 7,235,733 2004 互联网梅森素数大搜索,乔许·芬德利
M25964951 7,816,230 2005 互联网梅森素数大搜索,马田·诺或
M30402457 9,152,052 2005 互联网梅森素数大搜索柯蒂斯·库珀与史提夫·布恩
M32582657 9,808,358 2006 互联网梅森素数大搜索柯蒂斯·库珀与史提夫·布恩
M43112609 12,978,189 2008 互联网梅森素数大搜索,埃德森·史密夫
M57885161 17,425,170 2013 互联网梅森素数大搜索柯蒂斯·库珀
M74207281 22,338,618 2016 互联网梅森素数大搜索柯蒂斯·库珀
M77232917 23,249,425 2017 互联网梅森素数大搜索,强纳森·佩斯
M82589933 24,862,048 2018 互联网梅森素数大搜索,派翠克·拉罗次

互联网梅森素数大搜索发现了近十五个最大素数纪录。

二十大已知素数

克里斯·科德韦尔设有一列表,内共有已知最大的五千个素数[14][15],其中最大二十个列于下表。

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

参见

参考资料

  1. ^ 1.0 1.1 1.2 GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1. Mersenne Research, Inc. 2018-12-21 [2018-12-21]. (原始内容存档于2020-08-15). 
  2. ^ Caldwell, Chris. The largest known primes - Database Search Output. Prime Pages. [2018-06-03]. (原始内容存档于2021-03-12). 
  3. ^ 3.0 3.1 Caldwell, Chris. The Largest Known Prime by Year: A Brief History. Prime Pages. [2016-01-20]. (原始内容存档于2013-08-19). 
  4. ^ 最后一个非梅森素数为391,581 ⋅ 2216,193 − 1页面存档备份,存于互联网档案馆);参见The Largest Known Prime by Year: A Brief History页面存档备份,存于互联网档案馆),Caldwell着
  5. ^ Perfect Numbers. Penn State University. [2019-10-06]. (原始内容存档于2020-08-03). An interesting side note is about the binary representations of those numbers... 
  6. ^ 存档副本. [2020-07-30]. (原始内容存档于2020-08-15). 
  7. ^ 7.0 7.1 7.2 Record 12-Million-Digit Prime Number Nets $100,000 Prize. Electronic Frontier Foundation. 电子前哨基金会. 2009-10-14 [2011-11-26]. (原始内容存档于2011-08-05). 
  8. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize页面存档备份,存于互联网档案馆).
  9. ^ Best Inventions of 2008 - 29. The 46th Mersenne Prime. Time (时代公司). 2008-10-29 [2012-01-17]. (原始内容存档于2013-08-22). 
  10. ^ C. Edward Sandifer. How Euler Did Even More. 2007-08-30: 43 [2020-07-30]. ISBN 0883855844. (原始内容存档于2020-08-04). 
  11. ^ J. Miller英语J. C. P. Miller, Large Prime Numbers. Nature 168, 838 (1951).
  12. ^ Letters to the Editor页面存档备份,存于互联网档案馆). The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
  13. ^ Proof-code: Z页面存档备份,存于互联网档案馆), The Prime Pages英语Prime Pages.
  14. ^ The Prime Database: The List of Largest Known Primes Home Page. primes.utm.edu/primes. Chris K. Caldwell. [2017-09-30]. (原始内容存档于2021-02-27). 
  15. ^ The Top Twenty: Largest Known Primes. Chris K. Caldwell. [2018-01-03]. (原始内容存档于2021-02-25). 
  16. ^ GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1. mersenne.org. 互联网梅森素数大搜索. [2018-01-03]. (原始内容存档于2018-01-03). 
  17. ^ GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1. mersenne.org. 互联网梅森素数大搜索. [2017-09-29]. (原始内容存档于2018-01-07). 
  18. ^ GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2013-02-05 [2017-09-29]. (原始内容存档于2021-01-26). 
  19. ^ 19.0 19.1 GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2008-09-15 [2017-09-29]. (原始内容存档于2011-06-03). 
  20. ^ GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime.. mersenne.org. 互联网梅森素数大搜索. 2009-04-12 [2017-09-29]. (原始内容存档于2021-02-19). 
  21. ^ GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2006-09-11 [2017-09-29]. (原始内容存档于2021-01-26). 
  22. ^ PrimeGrid's Seventeen or Bust Subproject (PDF). primegrid.com. PrimeGrid英语PrimeGrid. [2017-09-30]. (原始内容存档 (PDF)于2021-01-15). 
  23. ^ GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2005-12-24 [2017-09-29]. (原始内容存档于2021-03-14). 
  24. ^ GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2005-02-27 [2017-09-29]. (原始内容存档于2021-03-14). 
  25. ^ GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2004-05-28 [2017-09-29]. (原始内容存档于2021-01-29). 
  26. ^ GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime.. mersenne.org. 互联网梅森素数大搜索. 2003-12-02 [2017-09-29]. (原始内容存档于2020-06-07). 
  27. ^ PrimeGrid's Generalized Fermat Prime Search (PDF). primegrid.com. PrimeGrid英语PrimeGrid. [2018-11-07]. (原始内容存档 (PDF)于2021-01-15). 
  28. ^ PrimeGrid's Generalized Fermat Prime Search (PDF). primegrid.com. PrimeGrid英语PrimeGrid. [2017-09-29]. (原始内容存档 (PDF)于2021-02-26). 
  29. ^ PrimeGrid's Prime Sierpinski Problem (PDF). primegrid.com. PrimeGrid英语PrimeGrid. [2017-09-29]. (原始内容存档 (PDF)于2020-12-23). 
  30. ^ The Prime Database: Phi(3,-123447^524288). primes.utm.edu. The Prime Pages. [2017-09-30]. (原始内容存档于2021-01-21). 
  31. ^ The Prime Database: 7*6^6772401+1. primes.utm.edu. The Prime Pages. [2019-09-12]. (原始内容存档于2021-01-21). 
  32. ^ PrimeGrid's Woodall Prime Search (PDF). primegrid.com. PrimeGrid英语PrimeGrid. [2018-04-02]. (原始内容存档 (PDF)于2021-01-21). 
  33. ^ The Prime Database: 6962*31^2863120-1. primes.utm.edu. The Prime Pages. [2020-04-06]. (原始内容存档于2021-01-21). 

外部链接