จำนวนเฉพาะแมร์แซน

จากวิกิพีเดีย สารานุกรมเสรี

จำนวนเฉพาะแมร์แซน (อังกฤษ: Mersenne prime) เป็นตัวเลขจำนวนเฉพาะที่อยู่ในรูปของ

จำนวนเฉพาะแมร์แซน ได้มาจากชื่อนักคณิตศาสตร์ชาวฝรั่งเศส มาแร็ง แมร์แซน (Marin Mersenne) มีชีวิตอยู่สมัยศตวรรษที่ 17 ได้รับการยกย่องว่าเป็นผู้คิดวิธีที่ง่ายที่สุดในการทดสอบเลขจำนวนเฉพาะ โดยได้ทำการศึกษาเลขจำนวนเฉพาะในรูปแบบ 2p - 1 ซึ่งพบว่า 2p - 1 ไม่เป็นจำนวนเฉพาะทุกตัว

จำนวนเฉพาะที่มากที่สุดเท่าที่มีการค้นพบ 274,207,281 − 1 เป็นจำนวนเฉพาะแมร์แซน ในลำดับที่ 49[1][2][3]

จำนวนเฉพาะที่มีขนาดใหญ่มาก (ใหญ่กว่า 10100) นำไปใช้ประโยชน์ในขั้นตอนวิธีเข้ารหัสลับแบบกุญแจสาธารณะ นอกจากนี้ยังใช้ในตารางแฮช (hash tables) และเครื่องสุ่มเลขเทียม

ตารางจำนวนเฉพาะ[แก้]

# p Mp Mp จำนวนหลักใน Mp เวลาที่ค้นพบ (ค.ศ.) ผู้ค้นพบ วิธี
1 2 3 1 c. 430 BC นักคณิตศาสตร์กรีกโบราณ[4]
2 3 7 1 c. 430 BC นักคณิตศาสตร์กรีกโบราณ[4]
3 5 31 2 c. 300 BC นักคณิตศาสตร์กรีกโบราณ[5]
4 7 127 3 c. 300 BC นักคณิตศาสตร์กรีกโบราณ[5]
5 13 8191 4 1456 คนทั่วไป[6][7] การหารเชิงทดลอง
6 17 131071 6 1588[8] Pietro Cataldi การหารเชิงทดลอง[9]
7 19 524287 6 1588 Pietro Cataldi การหารเชิงทดลอง[10]
8 31 2147483647 10 1772 Leonhard Euler[11][12] Enhanced trial division[13]
9 61 2305843009213693951 19 1883 November[14] I. M. Pervushin Lucas sequences
10 89 618970019642...137449562111 27 1911 June[15] Ralph Ernest Powers Lucas sequences
11 107 162259276829...578010288127 33 1914 June 1[16][17][18] Ralph Ernest Powers[19] Lucas sequences
12 127 170141183460...715884105727 39 1876 January 10[20] Édouard Lucas Lucas sequences
13 521 686479766013...291115057151 157 1952 January 30[21] Raphael M. Robinson LLT / SWAC
14 607 531137992816...219031728127 183 1952 January 30[21] Raphael M. Robinson LLT / SWAC
15 1,279 104079321946...703168729087 386 1952 June 25[22] Raphael M. Robinson LLT / SWAC
16 2,203 147597991521...686697771007 664 1952 October 7[23] Raphael M. Robinson LLT / SWAC
17 2,281 446087557183...418132836351 687 1952 October 9[23] Raphael M. Robinson LLT / SWAC
18 3,217 259117086013...362909315071 969 1957 September 8[24] Hans Riesel LLT / BESK
19 4,253 190797007524...815350484991 1,281 1961 November 3[25][26] Alexander Hurwitz LLT / IBM 7090
20 4,423 285542542228...902608580607 1,332 1961 November 3[25][26] Alexander Hurwitz LLT / IBM 7090
21 9,689 478220278805...826225754111 2,917 1963 May 11[27] Donald B. Gillies LLT / ILLIAC II
22 9,941 346088282490...883789463551 2,993 1963 May 16[27] Donald B. Gillies LLT / ILLIAC II
23 11,213 281411201369...087696392191 3,376 1963 June 2[27] Donald B. Gillies LLT / ILLIAC II
24 19,937 431542479738...030968041471 6,002 1971 March 4[28] Bryant Tuckerman LLT / IBM 360/91
25 21,701 448679166119...353511882751 6,533 1978 October 30[29] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174
26 23,209 402874115778...523779264511 6,987 1979 February 9[30] Landon Curt Noll LLT / CDC Cyber 174
27 44,497 854509824303...961011228671 13,395 1979 April 8[31][32] Harry L. Nelson & David Slowinski LLT / Cray 1
28 86,243 536927995502...709433438207 25,962 1982 September 25 David Slowinski LLT / Cray 1
29 110,503 521928313341...083465515007 33,265 1988 January 29[33][34] Walter Colquitt & Luke Welsh LLT / NEC SX-2[35]
30 132,049 512740276269...455730061311 39,751 1983 September 19[36] David Slowinski LLT / Cray X-MP
31 216,091 746093103064...103815528447 65,050 1985 September 1[37][38] David Slowinski LLT / Cray X-MP/24
32 756,839 174135906820...328544677887 227,832 1992 February 17 David Slowinski & Paul Gage LLT / Harwell Lab's Cray-2[39]
33 859,433 129498125604...243500142591 258,716 1994 January 4[40][41][42] David Slowinski & Paul Gage LLT / Cray C90
34 1,257,787 412245773621...976089366527 378,632 1996 September 3[43] David Slowinski & Paul Gage[44] LLT / Cray T94
35 1,398,269 814717564412...868451315711 420,921 1996 November 13 GIMPS / Joel Armengaud[45] LLT / Prime95 on 90 MHz Pentium
36 2,976,221 623340076248...743729201151 895,932 1997 August 24 GIMPS / Gordon Spence[46] LLT / Prime95 on 100 MHz Pentium
37 3,021,377 127411683030...973024694271 909,526 1998 January 27 GIMPS / Roland Clarkson[47] LLT / Prime95 on 200 MHz Pentium
38 6,972,593 437075744127...142924193791 2,098,960 1999 June 1 GIMPS / Nayan Hajratwala[48] LLT / Prime95 on 350 MHz Pentium II IBM Aptiva
39 13,466,917 924947738006...470256259071 4,053,946 2001 November 14 GIMPS / Michael Cameron[49] LLT / Prime95 on 800 MHz Athlon T-Bird
40 20,996,011 125976895450...762855682047 6,320,430 2003 November 17 GIMPS / Michael Shafer[50] LLT / Prime95 on 2 GHz Dell Dimension
41 24,036,583 299410429404...882733969407 7,235,733 2004 May 15 GIMPS / Josh Findley[51] LLT / Prime95 on 2.4 GHz Pentium 4
42 25,964,951 122164630061...280577077247 7,816,230 2005 February 18 GIMPS / Martin Nowak[52] LLT / Prime95 on 2.4 GHz Pentium 4
43 30,402,457 315416475618...411652943871 9,152,052 2005 December 15 GIMPS / Curtis Cooper & Steven Boone[53] LLT / Prime95 on 2 GHz Pentium 4
44 32,582,657 124575026015...154053967871 9,808,358 2006 September 4 GIMPS / Curtis Cooper & Steven Boone[54] LLT / Prime95 on 3 GHz Pentium 4
45 37,156,667 202254406890...022308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael Elvenich[55] LLT / Prime95 on 2.83 GHz Core 2 Duo
46[n 1] 42,643,801 169873516452...765562314751 12,837,064 2009 April 12[n 2] GIMPS / Odd M. Strindmo[56][n 3] LLT / Prime95 on 3 GHz Core 2
47[n 1] 43,112,609 316470269330...166697152511 12,978,189 2008 August 23 GIMPS / Edson Smith[55] LLT / Prime95 on Dell Optiplex 745
48[n 1] 57,885,161 581887266232...071724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper[57] LLT / Prime95 on 3 GHz Intel Core2 Duo E8400[58]
49[n 1] 74,207,281 300376418084...391086436351 22,338,618 2015 September 17[n 4] GIMPS / Curtis Cooper[1] LLT / Prime95 on Intel Core i7-4790

อ้างอิง[แก้]

  1. 1.0 1.1 Cooper, Curtis (7 January 2016). "Mersenne Prime Number discovery – 274207281 − 1 is Prime!". Mersenne Research, Inc. สืบค้นเมื่อ 22 January 2016. 
  2. Brook, Robert (January 19, 2016). "Prime number with 22 million digits is the biggest ever found". New Scientist. สืบค้นเมื่อ 19 January 2016. 
  3. Chang, Kenneth (21 January 2016). "New Biggest Prime Number = 2 to the 74 Mil ... Uh, It’s Big". New York Times. สืบค้นเมื่อ 22 January 2016. 
  4. 4.0 4.1 There is no mentioning among the ancient Egyptians of prime numbers, and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. "The Egyptians used ($) in the table above for the first primes r = 3, 5, 7, or 11 (also for r = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/n Table [Retrieved 2012-11-11]. In the school of Pythagoras (b. about 570 – d. about 495 BC) and the Pythagoreans, we find the first sure observations of prime numbers. Hence the first two Mersenne primes, 3 and 7, were known to and may even be said to have been discovered by them. There is no reference, though, to their special form 22 − 1 and 23 − 1 as such. The sources to the knowledge of prime numbers among the Pythagoreans are late. The Neoplatonic philosopher Iamblichus, AD c. 245–c. 325, states that the Greek Platonic philosopher Speusippus, c. 408 – 339/8 BC, wrote a book named On Pythagorean Numbers. According to Iamblichus this book was based on the works of the Pythagorean Philolaus, c. 470–c. 385 BC, who lived a century after Pythagoras, 570 – c. 495 BC. In his Theology of Arithmetic in the chapter On the Decad, Iamblichus writes: "Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book On Pythagorean Numbers. In the first half of the book, he elegantly expounds linear numbers [i.e. prime numbers], polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 112f. [Retrieved 2012-11-11]. Iamblichus also gives us a direct quote from Speusippus' book where Speusippus among other things writes: "Secondly, it is necessary for a perfect number [the concept "perfect number" is not used here in a modern sense] to contain an equal amount of prime and incomposite numbers, and secondary and composite numbers." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 113. [Retrieved 2012-11-11]. For the Greek original text, see Speusippus of Athens: A Critical Study with a Collection of the Related Texts and Commentary by Leonardo Tarán, 1981, p. 140 line 21–22 [Retrieved 2012-11-11] In his comments to Nicomachus of Gerasas's Introduction to Arithmetic, Iamblichus also mentions that Thymaridas, ca. 400 BC – ca. 350 BC, uses the term rectilinear for prime numbers, and that Theon of Smyrna, fl. AD 100, uses euthymetric and linear as alternative terms. Nicomachus of Gerasa, Introduction to Arithmetic, 1926, p. 127 [Retrieved 2012-11-11] It is unclear though when this said Thymaridas lived. "In a highly suspect passage in Iamblichus, Thymaridas is listed as a pupil of Pythagoras himself." Pythagoreanism [Retrieved 2012-11-11] Before Philolaus, c. 470–c. 385 BC, we have no proof of any knowledge of prime numbers.
  5. 5.0 5.1 "Euclid's Elements, Book IX, Proposition 36". 
  6. The Prime Pages, Mersenne Primes: History, Theorems and Lists.
  7. We find the oldest (undisputed) note of the result in Codex nr. 14908, which origins from Bibliotheca monasterii ord. S. Benedicti ad S. Emmeramum Ratisbonensis now in the archive of the Bayerische Staatsbibliothek, see "Halm, Karl / Laubmann, Georg von / Meyer, Wilhelm: Catalogus codicum latinorum Bibliothecae Regiae Monacensis, Bd.: 2,2, Monachii, 1876, p. 250". [retrieved on 2012-09-17] The Codex nr. 14908 consists of 10 different medieval works on mathematics and related subjects. The authors of most of these writings are known. Some authors consider the monk Fridericus Gerhart (Amman), 1400–1465 (Frater Fridericus Gerhart monachus ordinis sancti Benedicti astrologus professus in monasterio sancti Emmerani diocesis Ratisponensis et in ciuitate eiusdem) to be the author of the part where the prime number 8191 is mentioned. Geschichte Der Mathematik [retrieved on 2012-09-17] The second manuscript of Codex nr. 14908 has the name "Regulae et exempla arithmetica, algebraica, geometrica" and the 5th perfect number and all is factors, including 8191, are mentioned on folio no. 34 a tergo (backside of p. 34). Parts of the manuscript have been published in Archiv der Mathematik und Physik, 13 (1895), pp. 388–406 [retrieved on 2012-09-23]
  8. "A i lettori. Nel trattato de' numeri perfetti, che giàfino dell anno 1588 composi, oltrache se era passato auáti à trouarne molti auertite molte cose, se era anco amplamente dilatatala Tauola de' numeri composti , di ciascuno de' quali si vedeano per ordine li componenti, onde preposto unnum." p. 1 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  9. pp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  10. pp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  11. http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres 1772, pp. 35–36 EULER, Leonhard: Extrait d'une lettre à M. Bernoulli, concernant le Mémoire imprimé parmi ceux de 1771. p. 318 [intitulé: Recherches sur les diviseurs de quelques nombres très grands compris dans la somme de la progression géométrique 1 + 101 + 102 + 103 + ... + 10T = S]. Retrieved 2011-10-02.
  12. http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.
  13. Chris K. Caldwell. "Modular restrictions on Mersenne divisors". Primes.utm.edu. สืบค้นเมื่อ 2011-05-21. 
  14. “En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre 261 − 1 = 2305843009213693951 est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. http://www.biodiversitylibrary.org/item/107789#page/277/mode/1up [retrieved 2012-09-17] See also Mélanges mathématiques et astronomiques tirés du Bulletin de l’Académie impériale des sciences de St.-Pétersbourg v. 6 (1881–1888), pp. 553–554. See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48
  15. Powers, R. E. (1 January 1911). "The Tenth Perfect Number". The American Mathematical Monthly 18 (11): 195–197. JSTOR 2972574. doi:10.2307/2972574. 
  16. "M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1er Juin un cablógramme à M. Bromwich [secretary of London Mathematical Society] pour M107. Sur ma demande, ces deux auteurs m’ont adressé leurs remarquables résultats, et je m’empresse de les publier dans nos colonnes, avec nos felicitations." p. 103, André Gérardin, Nombres de Mersenne pp. 85, 103–108 in Sphinx-Œdipe. [Journal mensuel de la curiosité, de concours & de mathématiques.] v. 9, No. 1, 1914.
  17. "Power's cable announcing this same result was sent to the London Math. So. on 1 June 1914." Mersenne's Numbers, Scripta Mathematica, v. 3, 1935, pp. 112–119 http://primes.utm.edu/mersenne/LukeMirror/lit/lit_008s.htm [retrieved 2012-10-13]
  18. http://plms.oxfordjournals.org/content/s2-13/1/1.1.full.pdf Proceedings / London Mathematical Society (1914) s2–13 (1): 1. Result presented at a meeting with London Mathematical Society on June 11, 1914. Retrieved 2011-10-02.
  19. The Prime Pages, M107: Fauquembergue or Powers?.
  20. http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3039&I=166&M=chemindefer Presented at a meeting with Académie des sciences (France) on January 10, 1876. Retrieved 2011-10-02.
  21. 21.0 21.1 "Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2521 − 1 and 2607 − 1 on January 30, 1952." D. H. Lehmer, Recent Discoveries of Large Primes, Mathematics of Computation, vol. 6, No. 37 (1952), p. 61, http://www.ams.org/journals/mcom/1952-06-037/S0025-5718-52-99404-0/S0025-5718-52-99404-0.pdf [Retrieved 2012-09-18]
  22. "The program described in Note 131 (c) has produced the 15th Mersenne prime 21279 − 1 on June 25. The SWAC tests this number in 13 minutes and 25 seconds." D. H. Lehmer, A New Mersenne Prime, Mathematics of Computation, vol. 6, No. 39 (1952), p. 205, http://www.ams.org/journals/mcom/1952-06-039/S0025-5718-52-99387-3/S0025-5718-52-99387-3.pdf [Retrieved 2012-09-18]
  23. 23.0 23.1 "Two more Mersenne primes, 22203 − 1 and 22281 − 1, were discovered by the SWAC on October 7 and 9, 1952." D. H. Lehmer, Two New Mersenne Primes, Mathematics of Computation, vol. 7, No. 41 (1952), p. 72, http://www.ams.org/journals/mcom/1953-07-041/S0025-5718-53-99371-5/S0025-5718-53-99371-5.pdf [Retrieved 2012-09-18]
  24. "On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number M3217 = 23217 − 1 is a prime." Hans Riesel, A New Mersenne Prime, Mathematics of Computation, vol. 12 (1958), p. 60, http://www.ams.org/journals/mcom/1958-12-061/S0025-5718-1958-0099752-6/S0025-5718-1958-0099752-6.pdf [Retrieved 2012-09-18]
  25. 25.0 25.1 A. Hurwitz and J. L. Selfridge, Fermat numbers and perfect numbers, Notices of the American Mathematical Society, v. 8, 1961, p. 601, abstract 587-104.
  26. 26.0 26.1 "If p is prime, Mp = 2p − 1 is called a Mersenne number. The primes M4253 and M4423 were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18]
  27. 27.0 27.1 27.2 "The primes M9689, M9941, and M11213 which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois." Donald B. Gillies, Three New Mersenne Primes and a Statistical Theory, Mathematics of Computation, vol. 18, No. 85 (1964), pp. 93–97, http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0159774-6/S0025-5718-1964-0159774-6.pdf [Retrieved 2012-09-18]
  28. "On the evening of March 4, 1971, a zero Lucas-Lehmer residue for p = p24 = 19937 was found. Hence, M19937 is the 24th Mersenne prime." Bryant Tuckerman, The 24th Mersenne Prime, Proceedings of the National Academy of Sciences of the United States of America, vol. 68:10 (1971), pp. 2319–2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]
  29. "On October 30, 1978 at 9:40 pm, we found M21701 to be prime. The CPU time required for this test was 7:40:20. Tuckerman and Lehmer later provided confirmation of this result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
  30. "Of the 125 remaining Mp only M23209 was found to be prime. The test was completed on February 9, 1979 at 4:06 after 8:39:37 of CPU time. Lehmer and McGrogan later confirmed the result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
  31. David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013
  32. "The 27th Mersenne prime. It has 13395 digits and equals 244497 – 1. [...] Its primeness was determined on April 8, 1979 using the Lucas–Lehmer test. The test was programmed on a CRAY-1 computer by David Slowinski & Harry Nelson." (p. 15) "The result was that after applying the Lucas–Lehmer test to about a thousand numbers, the code determined, on Sunday, April 8th, that 244497 − 1 is, in fact, the 27th Mersenne prime." (p. 17), David Slowinski, "Searching for the 27th Mersenne Prime", Cray Channels, vol. 4, no. 1, (1982), pp. 15–17.
  33. "An FFT containing 8192 complex elements, which was the minimum size required to test M110503, ran approximately 11 minutes on the SX-2. The discovery of M110503 (January 29, 1988) has been confirmed." W. N. Colquitt and L. Welsh, Jr., A New Mersenne Prime, Mathematics of Computation, vol. 56, No. 194 (April 1991), pp. 867–870, http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068823-9/S0025-5718-1991-1068823-9.pdf [Retrieved 2012-09-18]
  34. "This week, two computer experts found the 31st Mersenne prime. But to their surprise, the newly discovered prime number falls between two previously known Mersenne primes. It occurs when p = 110,503, making it the third-largest Mersenne prime known." I. Peterson, Priming for a lucky strike Science News; 2/6/88, Vol. 133 Issue 6, pp. 85–85. http://ehis.ebscohost.com/ehost/detail?vid=3&hid=23&sid=9a9d7493-ffed-410b-9b59-b86c63a93bc4%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8824187 [Retrieved 2012-09-18]
  35. "Mersenne Prime Numbers". Omes.uni-bielefeld.de. 2011-01-05. สืบค้นเมื่อ 2011-05-21. 
  36. "Slowinski, a software engineer for Cray Research Inc. in Chippewa Falls, discovered the number at 11:36 a.m. Monday. [i.e. 1983 September 19]" Jim Higgins, "Elusive numeral's number is up" and "Scientist finds big number" in The Milwaukee Sentinel – Sep 24, 1983, p. 1, p. 11 [retrieved 2012-10-23]
  37. "The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form 2p − 1, where the exponent p is also a prime number. For instance, 127 is a Mersenne number for which the exponent is 7. The record prime number's exponent is 216,091." I. Peterson, Prime time for supercomputers Science News; 9/28/85, Vol. 128 Issue 13, p. 199. http://ehis.ebscohost.com/ehost/detail?vid=4&hid=22&sid=c11090a2-4670-469f-8f75-947b593a56a0%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8840537 [Retrieved 2012-09-18]
  38. "Slowinski's program also found the 28th in 1982, the 29th in 1983, and the 30th [known at that time] this past Labor Day weekend. [i.e. August 31-September 1, 1985]" Rad Sallee, "`Supercomputer'/Chevron calculating device finds a bigger prime number" Houston Chronicle, Friday 09/20/1985, Section 1, Page 26, 4 Star Edition [retrieved 2012-10-23]
  39. The Prime Pages, The finding of the 32nd Mersenne.
  40. Chris Caldwell, The Largest Known Primes.
  41. Crays press release
  42. "Slowinskis email". 
  43. Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
  44. The Prime Pages, A Prime of Record Size! 21257787 – 1.
  45. GIMPS Discovers 35th Mersenne Prime.
  46. GIMPS Discovers 36th Known Mersenne Prime.
  47. GIMPS Discovers 37th Known Mersenne Prime.
  48. GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
  49. GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
  50. GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
  51. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583 – 1.
  52. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951 – 1.
  53. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457 – 1.
  54. GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657 – 1.
  55. 55.0 55.1 Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
  56. "On April 12th [2009], the 47th known Mersenne prime, 242,643,801 – 1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.", The List of Largest Known Primes Home Page, http://primes.utm.edu/primes/page.php?id=88847 [retrieved 2012-09-18]
  57. "GIMPS Discovers 48th Mersenne Prime, 257,885,161 − 1 is now the Largest Known Prime.". Great Internet Mersenne Prime Search. สืบค้นเมื่อ 2016-01-19. 
  58. "List of known Mersenne prime numbers". สืบค้นเมื่อ 29 November 2014. 


อ้างอิงผิดพลาด: มีป้ายระบุ <ref> สำหรับกลุ่มชื่อ "n" แต่ไม่พบป้ายระบุ <references group="n"/> ที่สอดคล้องกัน หรือไม่มีการปิด </ref>