ดูต้นฉบับสำหรับ จำนวนเฉพาะแมร์แซน

'''จำนวนเฉพาะแมร์แซน''' ({{langx|en|Mersenne prime}}) เป็นตัวเลข[[จำนวนเฉพาะ]]ที่อยู่ในรูปของ

::<math>M_p = 2^p-1\,</math>

จำนวนเฉพาะแมร์แซน ได้มาจากชื่อ[[นักคณิตศาสตร์]]ชาว[[ฝรั่งเศส]] [[มาแร็ง แมร์แซน]] (Marin Mersenne) มีชีวิตอยู่สมัยศตวรรษที่ 17 ได้รับการยกย่องว่าเป็นผู้คิดวิธีที่ง่ายที่สุดในการทดสอบเลขจำนวนเฉพาะ โดยได้ทำการศึกษาเลขจำนวนเฉพาะในรูปแบบ 2<sup>''p''</sup> - 1 ซึ่งพบว่า 2<sup>''p''</sup> - 1 ไม่เป็นจำนวนเฉพาะทุกตัว 

[[จำนวนเฉพาะที่มากที่สุดเท่าที่มีการค้นพบ]] {{nowrap|2<sup>136,279,841</sup> − 1}} เป็นจำนวนเฉพาะแมร์แซน ในลำดับที่ 50

จำนวนเฉพาะที่มีขนาดใหญ่มาก (ใหญ่กว่า 10<sup>100</sup>) นำไปใช้ประโยชน์ในขั้นตอนวิธีเข้ารหัสลับแบบกุญแจสาธารณะ นอกจากนี้ยังใช้ใน[[ตารางแฮช]] (hash tables) และ[[เครื่องสุ่มเลขเทียม]]

== ตารางจำนวนเฉพาะ ==
{| class="wikitable"
|-
! #
! {{math|''p''}}
! {{math|''M<sub>p</sub>''}}
! {{math|''M<sub>p</sub>''}} จำนวนหลักใน Mp
! เวลาที่ค้นพบ (ค.ศ.)
! ผู้ค้นพบ
! วิธี
|-
| style="text-align:right;"| 1
| style="text-align:right;"| 2
| style="text-align:right;"| [[3 (number)|3]]
| style="text-align:right;"| 1
| c. 430 BC 
| [[Greek mathematics|นักคณิตศาสตร์กรีกโบราณ]]<ref name="Philolaus">There is no mentioning among the  [[Ancient Egypt|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 ($) in the table above for the first primes {{math|''r''}} = 3, 5, 7, or 11 (also for {{math|''r''}} = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptroll2-n.html The Rhind 2/{{math|''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 2<sup>2</sup>&nbsp;−&nbsp;1 and 2<sup>3</sup>&nbsp;−&nbsp;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." [https://www.scribd.com/doc/38568451/Theologoumena-Arithmeticae#page=112 ''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." [https://www.scribd.com/doc/38568451/Theologoumena-Arithmeticae#page=113 ''Iamblichus'' The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 113.] [Retrieved 2012-11-11]. For the Greek original text, see [https://books.google.com/books?id=cUPXqSb7V1wC&lpg=PA276&ots=Q3QhUOGtvH&dq=Speusippus%20prime&hl=sv&pg=PA140#v=onepage&q&f=false 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|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. [http://ia700709.us.archive.org//load_djvu_applet.php?file=27/items/NicomachusIntroToArithmetic/nicomachus_introduction_arithmetic.djvu 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."  [http://plato.stanford.edu/entries/pythagoreanism/#hippasus Pythagoreanism] [Retrieved 2012-11-11]

Before [[Philolaus]], c. 470–c. 385 BC, we have no proof of any knowledge of prime numbers.</ref>
|
|-
| style="text-align:right;"| 2
| style="text-align:right;"| 3
| style="text-align:right;"| [[7 (number)|7]]
| style="text-align:right;"| 1
| c. 430 BC
| นักคณิตศาสตร์กรีกโบราณ<ref name="Philolaus"/>
|
|-
| style="text-align:right;"| 3
| style="text-align:right;"| 5
| style="text-align:right;"| [[31 (number)|31]]
| style="text-align:right;"| 2
| c. 300 BC
| นักคณิตศาสตร์กรีกโบราณ<ref name="Euclid">{{cite web|url=http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html|title=Euclid's Elements, Book IX, Proposition 36|publisher=}}</ref>
|
|-
| style="text-align:right;"| 4
| style="text-align:right;"| 7
| style="text-align:right;"| [[127 (number)|127]]
| style="text-align:right;"| 3
| c. 300 BC
| นักคณิตศาสตร์กรีกโบราณ<ref name="Euclid"/>
|
|-
| style="text-align:right;"| 5
| style="text-align:right;"| 13
| style="text-align:right;"| [[8191 (number)|8191]]
| style="text-align:right;"| 4
| 1456
| คนทั่วไป<ref name="primepages">The Prime Pages, [http://primes.utm.edu/mersenne/ Mersenne Primes: History, Theorems and Lists].</ref><ref>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 / [http://daten.digitale-sammlungen.de/~db/bsb00008253/images/index.html?fip=193.174.98.30&seite=254 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. [http://audio02.archive.org/stream/vorlesungenber02cantuoft#page/238/mode/2up 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 [https://archive.org/stream/archivdermathem91unkngoog#page/n389/mode/2up Archiv der Mathematik und Physik, 13 (1895), pp. 388–406] [retrieved on 2012-09-23]</ref>
| [[การหารเชิงทดลอง]]
|-
| style="text-align:right;"| 6
| style="text-align:right;"| 17
| style="text-align:right;"| [[131071 (number)|131071]]
| style="text-align:right;"| 6
| 1588<ref>"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#{{ลิงก์เสีย|date=สิงหาคม 2021 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
| [[Pietro Cataldi]]
| การหารเชิงทดลอง<ref>pp. 13–18 in ''Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo'' 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#{{ลิงก์เสีย|date=สิงหาคม 2021 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
|-
| style="text-align:right;"| 7
| style="text-align:right;"| 19
| style="text-align:right;"| [[524287 (number)|524287]]
| style="text-align:right;"| 6
| 1588
| Pietro Cataldi
| การหารเชิงทดลอง<ref>pp. 18–22 in ''Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo'' 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#{{ลิงก์เสีย|date=สิงหาคม 2021 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>
|-
| style="text-align:right;"| 8
| style="text-align:right;"| 31
| style="text-align:right;"| [[2147483647 (number)|2147483647]]
| style="text-align:right;"| 10
| 1772
| [[Leonhard Euler]]<ref>http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 {{Webarchive|url=https://web.archive.org/web/20120331144007/http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv%2F1772&seite%3Aint=36 |date=2012-03-31 }} 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&nbsp;+&nbsp;101&nbsp;+&nbsp;102&nbsp;+&nbsp;103&nbsp;+&nbsp;...&nbsp;+&nbsp;10T&nbsp;=&nbsp;S].'' Retrieved 2011-10-02.</ref><ref>http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.</ref>
| Enhanced trial division<ref>{{cite web|author=Chris K. Caldwell |url=http://primes.utm.edu/notes/proofs/MerDiv.html |title=Modular restrictions on Mersenne divisors |publisher=Primes.utm.edu |date= |accessdate=2011-05-21}}</ref>
|-
| style="text-align:right;"| 9
| style="text-align:right;"| 61
| style="text-align:right;"| 2305843009213693951
| style="text-align:right;"| 19
| 1883 November<ref>“En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre

{{nowrap|2<sup>61</sup> − 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
</ref>
| [[Ivan Mikheevich Pervushin|I. M. Pervushin]]
| [[Lucas sequence]]s
|-
| style="text-align:right;"| 10
| style="text-align:right;"| 89
| style="text-align:right;"| 618970019642...137449562111
| style="text-align:right;"| 27
| 1911 June<ref>{{cite journal|jstor=2972574|title=The Tenth Perfect Number|first=R. E.|last=Powers|date=1 January 1911|publisher=|journal=The American Mathematical Monthly|volume=18|issue=11|pages=195–197|via=|doi=10.2307/2972574}}</ref>
| [[Ralph Ernest Powers]]
| Lucas sequences
|-
| style="text-align:right;"| 11
| style="text-align:right;"| 107
| style="text-align:right;"| 162259276829...578010288127
| style="text-align:right;"| 33
| 1914 June 1<ref>"M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1<sup>er</sup> Juin un cablógramme à [[Thomas John I'Anson Bromwich|M. Bromwich]] [secretary of London Mathematical Society] pour {{math|''M''<sub>107</sub>}}. 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.</ref><ref>"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]</ref><ref>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.</ref>
| Ralph Ernest Powers<ref>The Prime Pages, [http://primes.utm.edu/notes/fauquem.html {{math|''M''<sub>107</sub>}}: Fauquembergue or Powers?].</ref>
| Lucas sequences
|-
| style="text-align:right;"| 12
| style="text-align:right;"| 127
| style="text-align:right;"| 170141183460...715884105727
| style="text-align:right;"| 39
| 1876 January 10<ref>http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3039&I=166&M=chemindefer {{Webarchive|url=https://web.archive.org/web/20181006131732/http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3039&I=166&M=chemindefer |date=2018-10-06 }} Presented at a meeting with Académie des sciences (France) on January 10, 1876. Retrieved 2011-10-02.</ref>
| [[Édouard Lucas]]
| Lucas sequences
|-
| style="text-align:right;"| 13
| style="text-align:right;"| 521
| style="text-align:right;"| 686479766013...291115057151
| style="text-align:right;"| 157
| 1952 January 30<ref name="autogenerated4">"Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2<sup>521</sup>&nbsp;−&nbsp;1 and 2<sup>607</sup>&nbsp;−&nbsp;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]</ref>
| [[Raphael M. Robinson]]
| [[Lucas–Lehmer primality test|LLT]] / [[SWAC (computer)|SWAC]]
|-
| style="text-align:right;"| 14
| style="text-align:right;"| 607
| style="text-align:right;"| 531137992816...219031728127
| style="text-align:right;"| 183
| 1952 January 30<ref name="autogenerated4" />
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 15
| style="text-align:right;"| 1,279
| style="text-align:right;"| 104079321946...703168729087
| style="text-align:right;"| 386
| 1952 June 25<ref>"The program described in Note 131 (c) has produced the 15th Mersenne prime 2<sup>1279</sup>&nbsp;−&nbsp;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]</ref>
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 16
| style="text-align:right;"| 2,203
| style="text-align:right;"| 147597991521...686697771007
| style="text-align:right;"| 664
| 1952 October 7<ref name="autogenerated2">"Two more Mersenne primes, 2<sup>2203</sup>&nbsp;−&nbsp;1 and 2<sup>2281</sup>&nbsp;−&nbsp;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]</ref>
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 17
| style="text-align:right;"| 2,281
| style="text-align:right;"| 446087557183...418132836351
| style="text-align:right;"| 687
| 1952 October 9<ref name="autogenerated2" />
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 18
| style="text-align:right;"| 3,217
| style="text-align:right;"| 259117086013...362909315071
| style="text-align:right;"| 969
| 1957 September 8<ref>"On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number {{nowrap|{{math|''M''<sub>3217</sub>}} {{=}} 2<sup>3217</sup> − 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]</ref>
| [[Hans Riesel]]
| LLT / [[BESK]]
|-
| style="text-align:right;"| 19
| style="text-align:right;"| 4,253
| style="text-align:right;"| 190797007524...815350484991
| style="text-align:right;"| 1,281
| 1961 November 3<ref name="Hurwitz and Selfridge 1961">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.</ref><ref name="autogenerated5">"If {{math|''p''}} is prime, {{math|''M<sub>p</sub>'' {{=}} 2<sup>''p''</sup> − 1}} is called a Mersenne number. The primes {{math|''M''<sub>4253</sub>}} and {{math|''M''<sub>4423</sub>}} 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]</ref>
| Alexander Hurwitz
| LLT / [[IBM 7090]]
|-
| style="text-align:right;"| 20
| style="text-align:right;"| 4,423
| style="text-align:right;"| 285542542228...902608580607
| style="text-align:right;"| 1,332
| 1961 November 3<ref name="Hurwitz and Selfridge 1961" /><ref name="autogenerated5" />
| Alexander Hurwitz
| LLT / IBM 7090
|-
| style="text-align:right;"| 21
| style="text-align:right;"| 9,689
| style="text-align:right;"| 478220278805...826225754111
| style="text-align:right;"| 2,917
| 1963 May 11<ref name="autogenerated3">"The primes {{math|''M''<sub>9689</sub>}}, {{math|''M''<sub>9941</sub>}}, and {{math|''M''<sub>11213</sub>}} 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]</ref>
| [[Donald B. Gillies]]
| LLT / [[ILLIAC II]]
|-
| style="text-align:right;"| 22
| style="text-align:right;"| 9,941
| style="text-align:right;"| 346088282490...883789463551
| style="text-align:right;"| 2,993
| 1963 May 16<ref name="autogenerated3" />
| Donald B. Gillies
| LLT / ILLIAC II
|-
| style="text-align:right;"| 23
| style="text-align:right;"| 11,213
| style="text-align:right;"| 281411201369...087696392191
| style="text-align:right;"| 3,376
| 1963 June 2<ref name="autogenerated3" />
| Donald B. Gillies
| LLT / ILLIAC II
|-
| style="text-align:right;"| 24
| style="text-align:right;"| 19,937
| style="text-align:right;"| 431542479738...030968041471
| style="text-align:right;"| 6,002
| 1971 March 4<ref>"On the evening of March 4, 1971, a zero Lucas-Lehmer residue for {{math|''p'' {{=}} ''p''<sub>24</sub> {{=}} 19937}} was found. Hence, {{math|''M''<sub>19937</sub>}} 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]</ref>
| [[Bryant Tuckerman]]
| LLT / [[IBM 360]]/91
|-
| style="text-align:right;"| 25
| style="text-align:right;"| 21,701
| style="text-align:right;"| 448679166119...353511882751
| style="text-align:right;"| 6,533
| 1978 October 30<ref>"On October 30, 1978 at 9:40 pm, we found {{math|''M''<sub>21701</sub>}} 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]</ref>
| [[Landon Curt Noll]] & Laura Nickel
| LLT / [[CDC Cyber]] 174
|-
| style="text-align:right;"| 26
| style="text-align:right;"| 23,209
| style="text-align:right;"| 402874115778...523779264511
| style="text-align:right;"| 6,987
| 1979 February 9<ref>"Of the 125 remaining {{math|''M<sub>p</sub>''}} only {{math|''M''<sub>23209</sub>}} 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]</ref>
| Landon Curt Noll
| LLT / CDC Cyber 174
|-
| style="text-align:right;"| 27
| style="text-align:right;"| 44,497
| style="text-align:right;"| 854509824303...961011228671
| style="text-align:right;"| 13,395
| 1979 April 8<ref>David Slowinski, "Searching for the 27th Mersenne Prime", ''Journal of Recreational Mathematics'', v. 11(4), 1978–79, pp. 258–261, MR 80g #10013</ref><ref>"The 27th Mersenne prime. It has 13395 digits and equals 2<sup>44497</sup>&nbsp;–&nbsp;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 2<sup>44497</sup>&nbsp;−&nbsp;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.</ref>
| [[Harry L. Nelson]] & [[David Slowinski]]
| LLT / [[Cray 1]]
|-
| style="text-align:right;"| 28
| style="text-align:right;"| 86,243
| style="text-align:right;"| 536927995502...709433438207
| style="text-align:right;"| 25,962
| 1982 September 25
| David Slowinski
| LLT / Cray 1
|-
| style="text-align:right;"| 29
| style="text-align:right;"| 110,503
| style="text-align:right;"| 521928313341...083465515007
| style="text-align:right;"| 33,265
| 1988 January 29<ref>"An FFT containing 8192 complex elements, which was the minimum size required to test M<sub>110503</sub>, ran approximately 11 minutes on the SX-2. The discovery of {{math|''M''<sub>110503</sub>}} (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]</ref><ref>"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 {{math|''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]</ref>
| Walter Colquitt & Luke Welsh
| LLT / [[NEC SX architecture|NEC SX-2]]<ref>{{cite web|url=http://wwwhomes.uni-bielefeld.de/achim/mersenne.html |title=Mersenne Prime Numbers |publisher=Omes.uni-bielefeld.de |date=2011-01-05 |accessdate=2011-05-21}}</ref>
|-
| style="text-align:right;"| 30
| style="text-align:right;"| 132,049
| style="text-align:right;"| 512740276269...455730061311
| style="text-align:right;"| 39,751
| 1983 September 19<ref>"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]" [https://news.google.com/newspapers?id=roFQAAAAIBAJ&sjid=VRIEAAAAIBAJ&pg=1717%2C4841183 Jim Higgins, "Elusive numeral's number is up" and "Scientist finds big number" in ''The Milwaukee Sentinel'' – Sep 24, 1983, p. 1, ] [https://news.google.com/newspapers?id=roFQAAAAIBAJ&sjid=VRIEAAAAIBAJ&pg=4379%2C4887110 p. 11] [retrieved 2012-10-23]</ref>
| David Slowinski
| LLT / [[Cray X-MP]]
|-
| style="text-align:right;"| 31
| style="text-align:right;"| 216,091
| style="text-align:right;"| 746093103064...103815528447
| style="text-align:right;"| 65,050
| 1985 September 1<ref>"The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form {{math|2<sup>''p''</sup> − 1}}, where the exponent {{math|''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]</ref><ref>"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]" [http://www.chron.com/CDA/archives/archive.mpl/1985_39205/supercomputer-chevron-calculating-device-finds-a-b.html 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]</ref>
| David Slowinski
| LLT / Cray X-MP/24
|-
| style="text-align:right;"| 32
| style="text-align:right;"| 756,839
| style="text-align:right;"| 174135906820...328544677887
| style="text-align:right;"| 227,832
| 1992 February 17
| David Slowinski & [[Paul Gage]]
| LLT / [[Harwell Lab]]'s [[Cray-2]]<ref>The Prime Pages, [http://primes.utm.edu/notes/756839.html The finding of the 32''nd'' Mersenne].</ref>
|-
| style="text-align:right;"| 33
| style="text-align:right;"| 859,433
| style="text-align:right;"| 129498125604...243500142591
| style="text-align:right;"| 258,716
| 1994 January 4<ref>Chris Caldwell, [http://www.math.unicaen.fr/~reyssat/largest.html The Largest Known Primes] {{Webarchive|url=https://web.archive.org/web/19981202090207/http://www.math.unicaen.fr/%7Ereyssat/largest.html |date=1998-12-02 }}.</ref><ref>[http://www.0x07bell.net/WWWMASTER/CrayWWWStuff/prfoldercomp/PRIME_NUMBER.940110.txt Crays press release]</ref><ref>{{cite web|url=https://groups.google.com/forum/#!msg/sci.crypt/XXNkt07IE2s/eDeaWg5-mtIJ|title=Slowinskis email}}</ref>
| David Slowinski & Paul Gage
| LLT / [[Cray C90]]
|-
| style="text-align:right;"| 34
| style="text-align:right;"| 1,257,787
| style="text-align:right;"| 412245773621...976089366527
| style="text-align:right;"| 378,632
| 1996 September 3<ref>Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]</ref>
| David Slowinski & Paul Gage<ref>The Prime Pages, [http://primes.utm.edu/notes/1257787.html A Prime of Record Size! 2<sup>1257787</sup>&nbsp;–&nbsp;1].</ref>
| LLT / [[Cray T90|Cray T94]]
|-
| style="text-align:right;"| 35
| style="text-align:right;"| 1,398,269
| style="text-align:right;"| 814717564412...868451315711
| style="text-align:right;"| 420,921
| 1996 November 13
| [[Great Internet Mersenne Prime Search|GIMPS]] / Joel Armengaud<ref>[http://www.mersenne.org/primes/1398269.htm GIMPS Discovers 35th Mersenne Prime].</ref>
| LLT / [[Prime95]] on 90&nbsp;MHz [[Pentium]]
|-
| style="text-align:right;"| 36
| style="text-align:right;"| 2,976,221
| style="text-align:right;"| 623340076248...743729201151
| style="text-align:right;"| 895,932
| 1997 August 24
| GIMPS / Gordon Spence<ref>[http://www.mersenne.org/primes/2976221.htm GIMPS Discovers 36th Known Mersenne Prime].</ref>
| LLT / Prime95 on 100&nbsp;MHz Pentium
|-
| style="text-align:right;"| 37
| style="text-align:right;"| 3,021,377
| style="text-align:right;"| 127411683030...973024694271
| style="text-align:right;"| 909,526
| 1998 January 27
| GIMPS / Roland Clarkson<ref>[http://www.mersenne.org/primes/3021377.htm GIMPS Discovers 37th Known Mersenne Prime].</ref>
| LLT / Prime95 on 200&nbsp;MHz Pentium
|-
| style="text-align:right;"| 38
| style="text-align:right;"| 6,972,593
| style="text-align:right;"| 437075744127...142924193791
| style="text-align:right;"| 2,098,960
| 1999 June 1
| GIMPS / Nayan Hajratwala<ref>[http://www.mersenne.org/primes/6972593.htm GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award].</ref>
| LLT / Prime95 on 350&nbsp;MHz [[Pentium II]] [[IBM Aptiva]]
|-
| style="text-align:right;"| 39
| style="text-align:right;"| 13,466,917
| style="text-align:right;"| 924947738006...470256259071
| style="text-align:right;"| 4,053,946
| 2001 November 14
| GIMPS / Michael Cameron<ref>GIMPS, [http://www.mersenne.org/primes/13466917.htm Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid].</ref>
| LLT / Prime95 on 800&nbsp;MHz [[Athlon Thunderbird|Athlon T-Bird]]
|-
| style="text-align:right;"| 40
| style="text-align:right;"| 20,996,011
| style="text-align:right;"| 125976895450...762855682047
| style="text-align:right;"| 6,320,430
| 2003 November 17
| GIMPS / Michael Shafer<ref>GIMPS, [http://www.mersenne.org/primes/20996011.htm Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid].</ref>
| LLT / Prime95 on 2&nbsp;GHz [[Dell Dimension]]
|-
| style="text-align:right;"| 41
| style="text-align:right;"| 24,036,583
| style="text-align:right;"| 299410429404...882733969407
| style="text-align:right;"| 7,235,733
| 2004 May 15
| GIMPS / Josh Findley<ref>GIMPS, [http://www.mersenne.org/primes/24036583.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>24,036,583</sup>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2.4&nbsp;GHz [[Pentium 4]]
|-
| style="text-align:right;"| 42
| style="text-align:right;"| 25,964,951
| style="text-align:right;"| 122164630061...280577077247
| style="text-align:right;"| 7,816,230
| 2005 February 18
| GIMPS / Martin Nowak<ref>GIMPS, [http://www.mersenne.org/primes/25964951.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>25,964,951</sup>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2.4&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 43
| style="text-align:right;"| 30,402,457
| style="text-align:right;"| 315416475618...411652943871
| style="text-align:right;"| 9,152,052
| 2005 December 15
| GIMPS / [[Curtis Cooper (mathematician)|Curtis Cooper]] & Steven Boone<ref>GIMPS, [http://www.mersenne.org/primes/30402457.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>30,402,457</sup>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 44
| style="text-align:right;"| 32,582,657
| style="text-align:right;"| 124575026015...154053967871
| style="text-align:right;"| 9,808,358
| 2006 September 4
| GIMPS / Curtis Cooper & Steven Boone<ref>GIMPS, [http://www.mersenne.org/primes/32582657.htm Mersenne.org Project Discovers Largest Known Prime Number, 2<sup>32,582,657</sup>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 3&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 45
| style="text-align:right;"| 37,156,667
| style="text-align:right;"| 202254406890...022308220927
| style="text-align:right;"| 11,185,272
| 2008 September 6
| GIMPS / Hans-Michael Elvenich<ref name="mp">[http://mersenne.org/primes/m45and46.htm Titanic Primes Raced to Win $100,000 Research Award]. Retrieved on 2008-09-16.</ref>
| LLT / Prime95 on 2.83&nbsp;GHz [[Core 2 Duo]]
|-
| style="text-align:right;"| 46<ref group="n" name="unverified_index">It is not verified whether any undiscovered Mersenne primes exist between the 45th ({{math|''M''<sub>37,156,667</sub>}}) and the 49th ({{math|''M''<sub>74,207,281</sub>}}) on this chart; the ranking is therefore provisional.</ref>
| style="text-align:right;"| 42,643,801
| style="text-align:right;"| 169873516452...765562314751
| style="text-align:right;"| 12,837,064
| 2009 April 12<ref group="n" name="date_issue">{{math|''M''<sub>42,643,801</sub>}} 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.</ref>
| GIMPS / Odd M. Strindmo<ref>"On April 12th [2009], the 47th known Mersenne prime, 2<sup>42,643,801</sup>&nbsp;–&nbsp;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]</ref><ref group="n" name="Strindmo">Strindmo also uses the alias Stig M. Valstad.</ref>
| LLT / Prime95 on 3&nbsp;GHz Core 2
|-
| style="text-align:right;"| 47<ref group="n" name="unverified_index" />
| style="text-align:right;"| 43,112,609
| style="text-align:right;"| 316470269330...166697152511
| style="text-align:right;"| 12,978,189
| 2008 August 23
| GIMPS / Edson Smith<ref name="mp" />
| LLT / Prime95 on [[Dell Optiplex]] 745
|-
| style="text-align:right;"| 48<ref group="n" name="unverified_index" />
| style="text-align:right;"| 57,885,161
| style="text-align:right;"| 581887266232...071724285951
| style="text-align:right;"| 17,425,170
| 2013 January 25
| GIMPS / Curtis Cooper<ref name="m48">{{cite web|url=http://mersenne.org/various/57885161.htm|title=GIMPS Discovers 48th Mersenne Prime, 2<sup>57,885,161</sup> − 1 is now the Largest Known Prime.|publisher=[[Great Internet Mersenne Prime Search]]|accessdate=2016-01-19}}</ref>
| LLT / Prime95 on 3&nbsp;GHz Intel Core2 Duo E8400<ref>{{cite web|url=http://www.mersenne.org/primes/|title=List of known Mersenne prime numbers|accessdate=29 November 2014}}</ref>
|-
| style="text-align:right;"| 49<ref group="n" name="unverified_index" />
| style="text-align:right;"| 74,207,281
| style="text-align:right;"| 300376418084...391086436351
| style="text-align:right;"| 22,338,618
| 2015 September 17<ref group="n" name="date_issue_2>{{math|''M''<sub>74,207,281</sub>}} 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.</ref>
| GIMPS / Curtis Cooper<ref name="GIMPS-2016">{{cite web|last=Cooper|first=Curtis|date=7 January 2016|title=Mersenne Prime Number discovery – 2<sup>74207281</sup>&nbsp;−&nbsp;1 is Prime!|url=http://www.mersenne.org/primes/?press=M74207281|work=Mersenne Research, Inc.|accessdate=22 January 2016}}</ref>
| LLT / Prime95 on Intel [[Core i7-4790]]
|}

== เชิงอรรถ ==
=== หมายเหตุ ===
<references group="n"/>

=== อ้างอิง ===
{{รายการอ้างอิง}}
{{เริ่มอ้างอิง}}
* [http://www.manager.co.th/Science/ViewNews.aspx?NewsID=9490000000923 ม.มิสซูรีพบจำนวนเฉพาะขนาดใหญ่ที่สุด 9.1 ล้านหลัก] {{Webarchive|url=https://web.archive.org/web/20070630221934/http://www.manager.co.th/Science/ViewNews.aspx?NewsID=9490000000923 |date=2007-06-30 }} โดย ผู้จัดการออนไลน์ 5 มกราคม 2549 06:55 น.
{{จบอ้างอิง}}

[[หมวดหมู่:ลำดับจำนวนเต็ม]]
[[หมวดหมู่:จำนวนเฉพาะ]]
[[หมวดหมู่:ปัญหาทางคณิตศาสตร์ที่ยังแก้ไม่ได้]]
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