จำนวนเฉพาะแมร์แซน

จำนวนเฉพาะแมร์แซน () เป็นตัวเลขจำนวนเฉพาะที่อยู่ในรูปของ

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

จำนวนเฉพาะที่มากที่สุดเท่าที่มีการค้นพบ  เป็นจำนวนเฉพาะแมร์แซน ในลำดับที่ 50

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

 ตารางจำนวนเฉพาะ 
 #    จำนวนหลักใน Mp เวลาที่ค้นพบ (ค.ศ.) ผู้ค้นพบ วิธี 1 2 3 1 c. 430 BC  นักคณิตศาสตร์กรีกโบราณThere 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 ($) in the table above for the first primes  = 3, 5, 7, or 11 (also for  = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/ 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. 2 3 7 1 c. 430 BC นักคณิตศาสตร์กรีกโบราณ 3 5 31 2 c. 300 BC นักคณิตศาสตร์กรีกโบราณ 4 7 127 3 c. 300 BC นักคณิตศาสตร์กรีกโบราณ 5 13 8191 4 1456 คนทั่วไปThe Prime Pages, Mersenne Primes: History, Theorems and Lists.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] การหารเชิงทดลอง 6 17 131071 6 1588"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# Pietro Cataldi การหารเชิงทดลองpp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775# 7 19 524287 6 1588 Pietro Cataldi การหารเชิงทดลองpp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775# 8 31 2147483647 10 1772 Leonhard Eulerhttp://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.http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible. Enhanced trial division 9 61 2305843009213693951 19 1883 November“En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre

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 I. M. Pervushin Lucas sequences 10 89 618970019642...137449562111 27 1911 June Ralph Ernest Powers Lucas sequences 11 107 162259276829...578010288127 33 1914 June 1"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 . 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."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]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. Ralph Ernest PowersThe Prime Pages, : Fauquembergue or Powers?. Lucas sequences 12 127 170141183460...715884105727 39 1876 January 10http://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. Édouard Lucas Lucas sequences 13 521 686479766013...291115057151 157 1952 January 30"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] Raphael M. Robinson LLT / SWAC 14 607 531137992816...219031728127 183 1952 January 30 Raphael M. Robinson LLT / SWAC 15 1,279 104079321946...703168729087 386 1952 June 25"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] Raphael M. Robinson LLT / SWAC 16 2,203 147597991521...686697771007 664 1952 October 7"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] Raphael M. Robinson LLT / SWAC 17 2,281 446087557183...418132836351 687 1952 October 9 Raphael M. Robinson LLT / SWAC 18 3,217 259117086013...362909315071 969 1957 September 8"On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number  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] Hans Riesel LLT / BESK 19 4,253 190797007524...815350484991 1,281 1961 November 3A. Hurwitz and J. L. Selfridge, Fermat numbers and perfect numbers, Notices of the American Mathematical Society, v. 8, 1961, p. 601, abstract 587-104."If  is prime,  is called a Mersenne number. The primes  and  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] Alexander Hurwitz LLT / IBM 7090 20 4,423 285542542228...902608580607 1,332 1961 November 3 Alexander Hurwitz LLT / IBM 7090 21 9,689 478220278805...826225754111 2,917 1963 May 11"The primes , , and  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] Donald B. Gillies LLT / ILLIAC II 22 9,941 346088282490...883789463551 2,993 1963 May 16 Donald B. Gillies LLT / ILLIAC II 23 11,213 281411201369...087696392191 3,376 1963 June 2 Donald B. Gillies LLT / ILLIAC II 24 19,937 431542479738...030968041471 6,002 1971 March 4"On the evening of March 4, 1971, a zero Lucas-Lehmer residue for  was found. Hence,  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] Bryant Tuckerman LLT / IBM 360/91 25 21,701 448679166119...353511882751 6,533 1978 October 30"On October 30, 1978 at 9:40 pm, we found  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] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174 26 23,209 402874115778...523779264511 6,987 1979 February 9"Of the 125 remaining  only  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] Landon Curt Noll LLT / CDC Cyber 174 27 44,497 854509824303...961011228671 13,395 1979 April 8David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013"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. 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"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  (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]"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 , 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] Walter Colquitt & Luke Welsh LLT / NEC SX-2 30 132,049 512740276269...455730061311 39,751 1983 September 19"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] David Slowinski LLT / Cray X-MP 31 216,091 746093103064...103815528447 65,050 1985 September 1"The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form , where the exponent  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]"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] 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-2The Prime Pages, The finding of the 32nd Mersenne. 33 859,433 129498125604...243500142591 258,716 1994 January 4Chris Caldwell, The Largest Known Primes .Crays press release David Slowinski & Paul Gage LLT / Cray C90 34 1,257,787 412245773621...976089366527 378,632 1996 September 3Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20] David Slowinski & Paul GageThe Prime Pages, A Prime of Record Size! 21257787 – 1. LLT / Cray T94 35 1,398,269 814717564412...868451315711 420,921 1996 November 13 GIMPS / Joel ArmengaudGIMPS Discovers 35th Mersenne Prime. LLT / Prime95 on 90 MHz Pentium 36 2,976,221 623340076248...743729201151 895,932 1997 August 24 GIMPS / Gordon SpenceGIMPS Discovers 36th Known Mersenne Prime. LLT / Prime95 on 100 MHz Pentium 37 3,021,377 127411683030...973024694271 909,526 1998 January 27 GIMPS / Roland ClarksonGIMPS Discovers 37th Known Mersenne Prime. LLT / Prime95 on 200 MHz Pentium 38 6,972,593 437075744127...142924193791 2,098,960 1999 June 1 GIMPS / Nayan HajratwalaGIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award. LLT / Prime95 on 350 MHz Pentium II IBM Aptiva 39 13,466,917 924947738006...470256259071 4,053,946 2001 November 14 GIMPS / Michael CameronGIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid. LLT / Prime95 on 800 MHz Athlon T-Bird 40 20,996,011 125976895450...762855682047 6,320,430 2003 November 17 GIMPS / Michael ShaferGIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid. LLT / Prime95 on 2 GHz Dell Dimension 41 24,036,583 299410429404...882733969407 7,235,733 2004 May 15 GIMPS / Josh FindleyGIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583 – 1. LLT / Prime95 on 2.4 GHz Pentium 4 42 25,964,951 122164630061...280577077247 7,816,230 2005 February 18 GIMPS / Martin NowakGIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951 – 1. LLT / Prime95 on 2.4 GHz Pentium 4 43 30,402,457 315416475618...411652943871 9,152,052 2005 December 15 GIMPS / Curtis Cooper & Steven BooneGIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457 – 1. LLT / Prime95 on 2 GHz Pentium 4 44 32,582,657 124575026015...154053967871 9,808,358 2006 September 4 GIMPS / Curtis Cooper & Steven BooneGIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657 – 1. LLT / Prime95 on 3 GHz Pentium 4 45 37,156,667 202254406890...022308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael ElvenichTitanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16. LLT / Prime95 on 2.83 GHz Core 2 Duo 46It is not verified whether any undiscovered Mersenne primes exist between the 45th () and the 49th () on this chart; the ranking is therefore provisional. 42,643,801 169873516452...765562314751 12,837,064 2009 April 12 was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. GIMPS / Odd M. Strindmo"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]Strindmo also uses the alias Stig M. Valstad. LLT / Prime95 on 3 GHz Core 2 47 43,112,609 316470269330...166697152511 12,978,189 2008 August 23 GIMPS / Edson Smith LLT / Prime95 on Dell Optiplex 745 48 57,885,161 581887266232...071724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper LLT / Prime95 on 3 GHz Intel Core2 Duo E8400 49 74,207,281 300376418084...391086436351 22,338,618 2015 September 17 was first found by a machine on September 17, 2015; however, no human took notice of this fact until January 7, 2016. Thus, either date may be considered the 'discovery' date. GIMPS considers the January 2016 date to be the official one. GIMPS / Curtis Cooper LLT / Prime95 on Intel Core i7-4790

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