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梅森質數同完全數一覽

出自維基百科，自由嘅百科全書
Cuisenaire rods showing the proper divisors of 6 (1, 2, and 3) adding up to 6
6 係個完全數
A graph plotting years on the x-axis with the number of digits of the largest known prime logarithmically on the y-axis, with two trendlines
近年已知最大質數嘅位數，用對數尺度畫出嚟，大部份呢啲質數都係梅森質數。

梅森質數同完全數係數論入面兩組緊密相關嘅自然數。梅森質數以數學家馬蘭·梅森命名，係指可以寫做2p − 1嘅質數，其中p係正整數。例如3就係一個梅森質數，因為佢係一個質數而且可以寫做22 − 1。[1][2] 對應梅森質數嘅數字p一定係質數，但係調返轉質數p唔一定對到一個梅森質數，例如11就對唔到：211 − 1 = 2047 = 23 × 89。[3] 另一方面，完全數就係「等於佢嘅正真因子加埋」嘅自然數，真因子係指自己以外嘅因子，所以例如6就係一個完全數，因為6嘅真因子係1, 2同3，而1 + 2 + 3 = 6。[2][4]

梅森質數同偶完全數之間有一一對應嘅關係。呢個結果叫做歐幾里得-歐拉定理，係由歐幾里得證明咗一半，再由歐拉證明剩低嗰一半：雙數係完全數若且唯若佢可以寫做2p − 1 × (2p − 1)咁嘅樣，當中2p − 1係梅森質數，亦即係話，所有可以寫做咁嘅樣嘅數都係完全數，而且所有偶完全數都可以寫做咁嘅樣。例如當p = 2嘅時候，22 − 1 = 3係一個質數，22 − 1 × (22 − 1) = 2 × 3 = 6係一個完全數。[1][5][6]

目前，係咪存在無限個梅森質數同偶完全數仲係一個未解決嘅問題[2][6] 梅森質數出現嘅頻率就係所謂嘅Lenstra–Pomerance–Wagstaff 猜想，佢哋估少於x嘅梅森質數數量係(eγ / log 2) × log log x，呢度e 係歐拉數，γ係歐拉常數，log係自然對數[7][8][9] 目前亦都未知有無奇完全數嘅存在，不過有好多人搵咗好多如果奇完全數存在嘅話要符合嘅條件，例如佢最細都起碼 101500。[10]

以下係目前已知所有梅森質數同對應嘅完全數表，有埋對應嘅次方 p。截至2023年 (2023-唔見咗必要嘅參數 1=月份！)，發現咗51個梅森質數，最大嗰17個係由分佈式計算計劃Great Internet Mersenne Prime Search（縮寫 GIMPS）發現嘅。[2] 新啲嘅梅森質數係用 Lucas-Lehmer 質數判定法 (LLT)驗算嘅，LLT 係個針對梅森質數好有效率嘅質數判定法。[2]

下邊展示嘅排名係截至2022年 (2022-唔見咗必要嘅參數 1=月份！)已知嘅質數排名。雖然機會唔大，不過如果發現到新嘅但係細啲嘅梅森質數嘅話，排名就會改。根據GIMPS，截至2023年8月 (2023-8月)所有48名亦即係p = 57,885,161之前嘅次方已經檢查哂。[11] 發現日子同發現人係指發現梅森質數，而完全數就可以即刻根據歐幾里得-歐拉定理計出嚟。發現人寫住「GIMPS / 人名」嘅係指經GIMPS發現，個人名就係發現嗰部機嘅主人。靠後嘅質數太長，淨係展示頭6位同尾6位。

目前已知51個梅森質數同對應嘅完全數
排名 p 梅森質數 梅森質數位數 完全數 完全數位數 發現日子 發現人 發現方法 Ref.[12]
1 2 3 1 6 1 古代[a] 古希臘數學家已經知 無記錄 [13][14][15]
2 3 7 1 28 2 [13][14][15]
3 5 31 2 496 3 [13][14][15]
4 7 127 3 8128 4 [13][14][15]
5 13 8191 4 33550336 8 c. 1456[b] 佚名[c] 試除法 [14][15]
6 17 131071 6 8589869056 10 1588年[b] Pietro Cataldi [2][18]
7 19 524287 6 137438691328 12 [2][18]
8 31 2147483647 10 230584...952128 19 1772年 歐拉 有商餘限制嘅試除法 [19][20]
9 61 230584...693951 19 265845...842176 37 1883年11月 Ivan Pervushin 盧卡斯數列 [21]
10 89 618970...562111 27 191561...169216 54 1911年6月 Ralph Ernest Powers [22]
11 107 162259...288127 33 131640...728128 65 1914年6月1號 [23]
12 127 170141...105727 39 144740...152128 77 1876年1月10號 Édouard Lucas [24]
13 521 686479...057151 157 235627...646976 314 1952年1月30號 Raphael M. Robinson SWAC上面嘅LLT [25]
14 607 531137...728127 183 141053...328128 366 [25]
15 1,279 104079...729087 386 541625...291328 770 1952年6月25號 [26]
16 2,203 147597...771007 664 108925...782528 1,327 1952年10月7號 [27]
17 2,281 446087...836351 687 994970...915776 1,373 1952年10月9號 [27]
18 3,217 259117...315071 969 335708...525056 1,937 1957年9月8號 Hans Riesel BESK上面嘅LLT [28]
19 4,253 190797...484991 1,281 182017...377536 2,561 1961年11月3號 Alexander Hurwitz IBM 7090上面嘅LLT [29]
20 4,423 285542...580607 1,332 407672...534528 2,663 [29]
21 9,689 478220...754111 2,917 114347...577216 5,834 1963年5月11號 Donald B. Gillies ILLIAC II上面嘅LLT [30]
22 9,941 346088...463551 2,993 598885...496576 5,985 1963年5月16號 [30]
23 11,213 281411...392191 3,376 395961...086336 6,751 1963年6月2號 [30]
24 19,937 431542...041471 6,002 931144...942656 12,003 1971年3月4號 Bryant Tuckerman IBM 360/91上面嘅LLT [31]
25 21,701 448679...882751 6,533 100656...605376 13,066 1978年10月30號 Landon Curt Noll & Laura Nickel CDC Cyber 174上面嘅LLT [32]
26 23,209 402874...264511 6,987 811537...666816 13,973 1979年2月9號 Landon Curt Noll [32]
27 44,497 854509...228671 13,395 365093...827456 26,790 1979年4月8號 Harry L. Nelson & David Slowinski Cray-1上面嘅LLT [33][34]
28 86,243 536927...438207 25,962 144145...406528 51,924 1982年9月25號 David Slowinski [35]
29 110,503 521928...515007 33,265 136204...862528 66,530 1988年1月29號 Walter Colquitt & Luke Welsh NEC SX-2上面嘅LLT [36][37]
30 132,049 512740...061311 39,751 131451...550016 79,502 1983年9月19號 David Slowinski et al. (Cray) Cray X-MP上面嘅LLT [38]
31 216,091 746093...528447 65,050 278327...880128 130,100 1985年9月1號 Cray X-MP/24上面嘅LLT [39][40]
32 756,839 174135...677887 227,832 151616...731328 455,663 1992年2月17號 Harwell Lab's Cray-2上面嘅LLT [41]
33 859,433 129498...142591 258,716 838488...167936 517,430 1994年1月4號 Cray C90上面嘅LLT [42]
34 1,257,787 412245...366527 378,632 849732...704128 757,263 1996年9月3號 Cray T94上面嘅LLT [43][44]
35 1,398,269 814717...315711 420,921 331882...375616 841,842 1996年11月13號 GIMPS / Joel Armengaud 90 MHz Pentium PC上面嘅LLT / Prime95 [45]
36 2,976,221 623340...201151 895,932 194276...462976 1,791,864 1997年8月24號 GIMPS / Gordon Spence 100 MHz Pentium PC上面嘅LLT / Prime95 [46]
37 3,021,377 127411...694271 909,526 811686...457856 1,819,050 1998年1月27號 GIMPS / Roland Clarkson 200 MHz Pentium PC上面嘅LLT / Prime95 [47]
38 6,972,593 437075...193791 2,098,960 955176...572736 4,197,919 1999年6月1號 GIMPS / Nayan Hajratwala IBM Aptiva with 350 MHz Pentium II processor上面嘅LLT / Prime95 [48]
39 13,466,917 924947...259071 4,053,946 427764...021056 8,107,892 2001年11月14號 GIMPS / Michael Cameron PC with 800 MHz Athlon T-Bird processor上面嘅LLT / Prime95 [49]
40 20,996,011 125976...682047 6,320,430 793508...896128 12,640,858 2003年11月17號 GIMPS / Michael Shafer Dell Dimension PC with 2 GHz Pentium 4 processor上面嘅LLT / Prime95 [50]
41 24,036,583 299410...969407 7,235,733 448233...950528 14,471,465 2004年5月15號 GIMPS / Josh Findley PC with 2.4 GHz Pentium 4 processor上面嘅LLT / Prime95 [51]
42 25,964,951 122164...077247 7,816,230 746209...088128 15,632,458 2005年2月18號 GIMPS / Martin Nowak [52]
43 30,402,457 315416...943871 9,152,052 497437...704256 18,304,103 2005年12月15號 GIMPS / Curtis Cooper & Steven Boone 喺中央密蘇里州立大學入面嘅 PC 上面嘅LLT / Prime95 [53]
44 32,582,657 124575...967871 9,808,358 775946...120256 19,616,714 2006年9月4號 [54]
45 37,156,667 202254...220927 11,185,272 204534...480128 22,370,543 2008年9月6號 GIMPS / Hans-Michael Elvenich PC上面嘅LLT / Prime95 [55]
46 42,643,801 169873...314751 12,837,064 144285...253376 25,674,127 2009年6月4號[d] GIMPS / Odd Magnar Strindmo PC（3 GHz Intel Core 2 處理器）上面嘅LLT / Prime95 [56]
47 43,112,609 316470...152511 12,978,189 500767...378816 25,956,377 2008年8月23號 GIMPS / Edson Smith Dell OptiPlex PC（Intel Core 2 Duo E6600處理器）上面嘅LLT / Prime95 [55][57][58]
48 57,885,161 581887...285951 17,425,170 169296...130176 34,850,340 2013年1月25號 GIMPS / Curtis Cooper 喺中央密蘇里州立大學入面嘅 PC 上面嘅LLT / Prime95 [59][60]
* 66,618,073 最低未驗證里程碑[e]
49[f] 74,207,281 300376...436351 22,338,618 451129...315776 44,677,235 2016年1月7號[g] GIMPS / Curtis Cooper PC（Intel Core i7-4790 處理器）上面嘅LLT / Prime95 [61][62]
50[f] 77,232,917 467333...179071 23,249,425 109200...301056 46,498,850 2017年12月26號 GIMPS / Jonathan Pace PC（Intel Core i5-6600 處理器）上面嘅LLT / Prime95 [63][64]
51[f] 82,589,933 148894...902591 24,862,048 110847...207936 49,724,095 2018年12月7號 GIMPS / Patrick Laroche PC（Intel Core i5-4590T 處理器）上面嘅LLT / Prime95 [65][66]
* 114,731,581 最低未測試里程碑[e]

喺過往紀錄入面，最大已知質數好多時都係梅森質數。

註

[編輯]
  1. ↑ The first four perfect numbers were documented by Nicomachus circa 100, and the concept was known (along with corresponding Mersenne primes) to Euclid at the time of his Elements. There is no record of discovery.
  2. ↑ 2.0 2.1 Islamic mathematicians such as Ismail ibn Ibrahim ibn Fallus (1194–1239) may have known of the fifth through seventh perfect numbers prior to European records.[16]
  3. ↑ Found in an anonymous manuscript, Clm 14908, dated 1456 and 1461[14][17]
  4. ↑ M42,643,801 was first reported to GIMPS on April 12, 2009 but was not noticed by a human until June 4, 2009 due to a server error.
  5. ↑ 5.0 5.1 截至2024年1月17號 (2024-1月-17)[11]
  6. ↑ 6.0 6.1 6.2 It has not been verified whether any undiscovered Mersenne primes exist between the 48th (M57,885,161) and the 51st (M82,589,933) on this table; the ranking is therefore provisional.
  7. ↑ M74,207,281 was first reported to GIMPS on September 17, 2015 but was not noticed by a human until January 7, 2016 due to a server error.

參考資料

[編輯]
  1. ↑ 1.0 1.1 Stillwell, John (2010). Mathematics and Its History. Undergraduate Texts in Mathematics. Springer Science+Business Media. p. 40. ISBN 978-1-4419-6052-8. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Caldwell, Chris K. "Mersenne Primes: History, Theorems and Lists". PrimePages. 原先內容歸檔喺4 October 2021. 喺4 October 2021搵到.
  3. ↑ Caldwell, Chris K. "If 2n-1 is prime, then so is n". PrimePages. 原先內容歸檔喺5 October 2021. 喺12 October 2021搵到.
  4. ↑ Prielipp, Robert W. (1970). "Perfect Numbers, Abundant Numbers, and Deficient Numbers". The Mathematics Teacher. 63 (8): 692–96. doi:10.5951/MT.63.8.0692. JSTOR 27958492. 原先內容歸檔喺5 October 2021. 喺13 October 2021搵到 –透過JSTOR.
  5. ↑ Caldwell, Chris K. "Characterizing all even perfect numbers". PrimePages. 原先內容歸檔喺8 October 2014. 喺12 October 2021搵到.
  6. ↑ 6.0 6.1 Crilly, Tony (2007). "Perfect numbers". 50 mathematical ideas you really need to know. Quercus Publishing. ISBN 978-1-84724-008-8. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到.
  7. ↑ Caldwell, Chris K. "Heuristics Model for the Distribution of Mersennes". PrimePages. 原先內容歸檔喺5 October 2021. 喺13 October 2021搵到.
  8. ↑ Wagstaff, Samuel S. (January 1983). "Divisors of Mersenne numbers". Mathematics of Computation (英文). 40 (161): 385–397. doi:10.1090/S0025-5718-1983-0679454-X. ISSN 0025-5718.
  9. ↑ Pomerance, Carl (September 1981). "Recent developments in primality testing" (PDF). The Mathematical Intelligencer (英文). 3 (3): 97–105. doi:10.1007/BF03022861. ISSN 0343-6993. S2CID 121750836.
  10. ↑ Ochem, Pascal; Rao, Michaël (30 January 2012). "Odd perfect numbers are greater than 101500". Mathematics of Computation (英文). 81 (279): 1869–1877. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718.
  11. ↑ 11.0 11.1 "GIMPS Milestones Report". Great Internet Mersenne Prime Search. 原先內容歸檔喺13 October 2021. 喺19 August 2023搵到.
  12. ↑ 一般參考：
  13. ↑ 13.0 13.1 13.2 13.3 Joyce, David E. "Euclid's Elements, Book IX, Proposition 36". mathcs.clarku.edu. 原先內容歸檔喺17 June 2021. 喺13 October 2021搵到.
  14. ↑ 14.0 14.1 14.2 14.3 14.4 14.5 Dickson, Leonard Eugene (1919). History of the Theory of Numbers, Vol. I. Carnegie Institution of Washington. pp. 4–6.
  15. ↑ 15.0 15.1 15.2 15.3 15.4 Smith, David Eugene (1925). History of Mathematics: Volume II. Dover. p. 21. ISBN 978-0-486-20430-7.
  16. ↑ O'Connor, John J.; Robertson, Edmund F. "Perfect numbers". MacTutor History of Mathematics archive. 原先內容歸檔喺5 October 2021. 喺13 October 2021搵到.
  17. ↑ "'Calendarium ecclesiasticum – BSB Clm 14908'". Bavarian State Library. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到.
  18. ↑ 18.0 18.1 Cataldi, Pietro Antonio (1603). Trattato de' numeri perfetti di Pietro Antonio Cataldo [Pietro Antonio Cataldi's treatise on perfect numbers] (義大利文). Presso di Heredi di Giouanni Rossi.
  19. ↑ Caldwell, Chris K. "Modular restrictions on Mersenne divisors". PrimePages. 喺22 November 2021搵到.
  20. ↑ Euler, Leonhard (1772). "Extrait d'un lettre de M. Euler le pere à M. Bernoulli concernant le Mémoire imprimé parmi ceux de 1771, p 318" [Extract of a letter from Mr. Euler to Mr. Bernoulli, concerning the Mémoire published among those of 1771]. Nouveaux Mémoires de l'académie royale des sciences de Berlin (法文). 1772: 35–36. 原先內容歸檔喺15 August 2020. 喺13 October 2021搵到 –透過Euler Archive.
  21. ↑ "Sur un nouveau nombre premier, annoncé par le père Pervouchine" [On a new prime number, announced by Pervouchine]. Bulletin de l'Académie impériale des sciences de St.-Pétersbourg (法文). 31: 532–533. 27 January 1887. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到 –透過Biodiversity Heritage Library.
  22. ↑ Powers, R. E. (November 1911). "The Tenth Perfect Number". The American Mathematical Monthly. 18 (11): 195–197. doi:10.2307/2972574. JSTOR 2972574.
  23. ↑ "Records of Proceedings at Meetings". Proceedings of the London Mathematical Society. s2-13 (1): iv–xl. 1914. doi:10.1112/plms/s2-13.1.1-s.
  24. ↑ Lucas, Édouard (1876). "Note sur l'application des séries récurrentes à la recherche de la loi de distribution des nombres premiers" [Note on the application of recurrent series to researching the law of prime number distribution]. Comptes rendus de l'Académie des Sciences (法文). 82: 165–167. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到.
  25. ↑ 25.0 25.1 "Notes". Mathematics of Computation (英文). 6 (37): 58–61. January 1952. doi:10.1090/S0025-5718-52-99405-2. ISSN 0025-5718. 原先內容歸檔喺13 October 2021. 喺13 October 2021搵到.
  26. ↑ "Notes". Mathematics of Computation (英文). 6 (39): 204–205. July 1952. doi:10.1090/S0025-5718-52-99389-7. ISSN 0025-5718.
  27. ↑ 27.0 27.1 "Notes". Mathematics of Computation (英文). 7 (41): 67–72. January 1953. doi:10.1090/S0025-5718-53-99372-7. ISSN 0025-5718.
  28. ↑ Riesel, Hans (January 1958). "A New Mersenne Prime". Mathematics of Computation. 12 (61): 60. doi:10.1090/S0025-5718-58-99282-2.
  29. ↑ 29.0 29.1 Hurwitz, Alexander (April 1962). "New Mersenne primes". Mathematics of Computation (英文). 16 (78): 249–251. doi:10.1090/S0025-5718-1962-0146162-X. ISSN 0025-5718.
  30. ↑ 30.0 30.1 30.2 Gillies, Donald B. (January 1964). "Three new Mersenne primes and a statistical theory". Mathematics of Computation. 18 (85): 93–97. doi:10.1090/S0025-5718-1964-0159774-6. JSTOR 2003409.
  31. ↑ Tuckerman, Bryant (October 1971). "The 24th Mersenne Prime". Proceedings of the National Academy of Sciences. 68 (10): 2319–2320. Bibcode:1971PNAS...68.2319T. doi:10.1073/pnas.68.10.2319. PMC 389411. PMID 16591945.
  32. ↑ 32.0 32.1 Noll, Landon Curt; Nickel, Laura (October 1980). "The 25th and 26th Mersenne primes". Mathematics of Computation. 35 (152): 1387. doi:10.1090/S0025-5718-1980-0583517-4. JSTOR 2006405.
  33. ↑ Slowinski, David (1978). "Searching for the 27th Mersenne prime". Journal of Recreational Mathematics. 11 (4): 258–261.
  34. ↑ "Science Watch: A New Prime Number". The New York Times. 5 June 1979. 喺13 October 2021搵到.
  35. ↑ "Announcements". The Mathematical Intelligencer (英文). 5 (1): 60. March 1983. doi:10.1007/BF03023507. ISSN 0343-6993.
  36. ↑ Peterson, I. (6 February 1988). "Priming for a Lucky Strike". Science News. 133 (6): 85. doi:10.2307/3972461. JSTOR 3972461.
  37. ↑ Colquitt, W. N.; Welsh, L. (April 1991). "A new Mersenne prime". Mathematics of Computation. 56 (194): 867. Bibcode:1991MaCom..56..867C. doi:10.1090/S0025-5718-1991-1068823-9. JSTOR 2008415.
  38. ↑ "Number is largest prime found yet". The Globe and Mail. 24 September 1983. Template:ProQuest –透過ProQuest.
  39. ↑ Peterson, I. (28 September 1985). "Prime Time for Supercomputers". Science News. 128 (13): 199. doi:10.2307/3970245. JSTOR 3970245.
  40. ↑ Dembart, Lee (17 September 1985). "Supercomputer Comes Up With Whopping Prime Number". Los Angeles Times (美國英文). 喺13 October 2021搵到.
  41. ↑ Maddox, John (26 March 1992). "The endless search for primality". Nature (英文). 356 (6367): 283. Bibcode:1992Natur.356..283M. doi:10.1038/356283a0. ISSN 1476-4687. S2CID 4327045.
  42. ↑ "Largest Known Prime Number Discovered on Cray Research Supercomputer". PR Newswire. 10 January 1994 –透過Gale.
  43. ↑ Caldwell, Chris K. "A Prime of Record Size! 21257787-1". PrimePages. 原先內容歸檔喺5 October 2021. 喺13 October 2021搵到.
  44. ↑ Gillmor, Dan (3 September 1996). "Crunching numbers: Researchers come up with prime math discovery". Knight Ridder –透過Gale.
  45. ↑ "GIMPS Discovers 35th Mersenne Prime, 21,398,269-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 12 November 1996. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  46. ↑ "GIMPS Discovers 36th Mersenne Prime, 22,976,221-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 1 September 1997. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  47. ↑ "GIMPS Discovers 37th Mersenne Prime, 23,021,377-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 2 February 1998. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  48. ↑ "GIMPS Discovers 38th Mersenne Prime 26,972,593-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 30 June 1999. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  49. ↑ "GIMPS Discovers 39th Mersenne Prime, 213,466,917-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 6 December 2001. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  50. ↑ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 2 February 2003. 原先內容歸檔喺7 June 2020. 喺13 October 2021搵到.
  51. ↑ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 28 May 2004. 原先內容歸檔喺29 January 2021. 喺13 October 2021搵到.
  52. ↑ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 27 February 2005. 原先內容歸檔喺14 March 2021. 喺13 October 2021搵到.
  53. ↑ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 24 December 2005. 原先內容歸檔喺14 March 2021. 喺13 October 2021搵到.
  54. ↑ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 11 September 2006. 原先內容歸檔喺26 January 2021. 喺13 October 2021搵到.
  55. ↑ 55.0 55.1 "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 15 September 2008. 原先內容歸檔喺5 October 2021. 喺13 October 2021搵到.
  56. ↑ "GIMPS Discovers 47th Mersenne Prime". Great Internet Mersenne Prime Search. 12 April 2009. 原先內容歸檔喺19 February 2021. 喺13 October 2021搵到.
  57. ↑ Maugh, Thomas H. (27 September 2008). "Rare prime number found". Los Angeles Times. 原先內容歸檔喺27 July 2021. 喺13 October 2021搵到.
  58. ↑ Smith, Edson. "The UCLA Mersenne Prime". UCLA Mathematics. 原著喺2021年11月22號歸檔. 喺22 November 2021搵到.
  59. ↑ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". Great Internet Mersenne Prime Search. 5 February 2013. 原先內容歸檔喺26 January 2021. 喺13 October 2021搵到.
  60. ↑ Yirka, Bob (6 February 2013). "University professor discovers largest prime number to date". phys.org (英文). 原先內容歸檔喺16 January 2021. 喺13 October 2021搵到.
  61. ↑ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". Great Internet Mersenne Prime Search. 19 January 2016. 原先內容歸檔喺7 January 2018. 喺13 October 2021搵到.
  62. ↑ "Largest known prime number discovered in Missouri". BBC News (英國英文). 20 January 2016. 原先內容歸檔喺21 August 2021. 喺13 October 2021搵到.
  63. ↑ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". Great Internet Mersenne Prime Search. 3 January 2018. 原先內容歸檔喺4 January 2018. 喺13 October 2021搵到.
  64. ↑ Lamb, Evelyn (4 January 2018). "Why You Should Care About a Prime Number That's 23,249,425 Digits Long". Slate Magazine (英文). 原先內容歸檔喺9 October 2021. 喺13 October 2021搵到.
  65. ↑ "GIMPS Discovers Largest Known Prime Number: 282,589,933-1". Great Internet Mersenne Prime Search. 21 December 2018. 原先內容歸檔喺22 December 2018. 喺13 October 2021搵到.
  66. ↑ Palca, Joe (21 December 2018). "The World Has A New Largest-Known Prime Number". NPR (英文). 原先內容歸檔喺30 July 2021. 喺13 October 2021搵到.

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