Mersenne-prímek forrásának megtekintése

{{megoldatlan|A|matematika|Létezik-e végtelen sok Mersenne-prím?}}
A [[matematika|matematikában]] '''Mersenne-prím'''eknek nevezzük a kettő-hatványnál eggyel kisebb, azaz a 2<sup>n</sup> ‒ 1 alakban felírható [[prímszámok]]at, ahol ''n'' szintén prímszám. A nevüket [[Marin Mersenne]] (1588–1648) francia szerzetes, matematikus, fizikus után kapták.

== Matematikai alapok ==

Például a 31 (prímszám) = 32 ‒ 1 = 2<sup>5</sup> ‒ 1, és 5 szintén prím, ezért a 31 egy Mersenne-prím; hasonlóan, 7 = 8 ‒ 1 = 2<sup>3</sup> ‒ 1. Másrészt 2047 = 2048 ‒ 1 = 2<sup>11</sup> ‒ 1, nem Mersenne-prím, mivel bár a 11 prímszám, a 2047 nem az (osztható 89-cel és 23-mal). 1952-től a legnagyobb ismert prímszám Mersenne-prím, kivéve az 1989–1992 közötti időszakot.<ref>The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "[http://primes.utm.edu/notes/by_year.html The Largest Known Prime by Year: A Brief History]" from the [[Prime Pages]] website, [[University of Tennessee at Martin]].</ref>

A Mersenne-prím definíciójában a kikötés, hogy ''n'' szükségképpen prím, elhagyható, ugyanis minden összetett ''n'' esetén elemi módon felbontható:

:<math>\begin{align}2^{ab}-1&=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)\\&=(2^b-1)\cdot \left(1+2^b+2^{2b}+2^{3b}+\cdots+2^{(a-1)b}\right).\end{align}</math>

Általánosabban, a '''Mersenne-számok''' (nem feltétlenül prímek, de lehetnek azok is) olyan [[természetes számok]], amelyek eggyel kisebbek egy kettő-hatványnál, tehát ''M<sub>n</sub>'' = 2<sup>''n''</sup> − 1. (A legtöbb forrás a Mersenne-számoknál is megköveteli, hogy az ''n'' prímszám legyen.)

== A Mersenne-prímek listája ==
[[Image:Primes.png|thumb|right|450px|Ábra az adott évben ismert legnagyobb Mersenne-prím számjegyeinek számáról]]
<!--Ha frissíted ezt a szakaszt, javítsd ezeket is: Prímszámok#A legnagyobb ismert prím és Prímszámok listája#Mersenne-prímek !!-->

{| class="wikitable"
|-
! Sorszám
! Hatványkitevő ({{math|''p''}})
! Mersenne-prím ({{math|''M<sub>p</sub>''}})
! Számjegy ({{math|''M<sub>p</sub>''}})
! Felfedezés éve
! Felfedező
! Használt módszer
|-
| style="text-align:right;"| 1
| style="text-align:right;"| 2
| style="text-align:right;"| [[3 (szám)|3]]
| style="text-align:right;"| 1
| i. e. 430 körül
| [[Görög matematika|Ókori görög matematikusok]]
|
|-
| style="text-align:right;"| 2
| style="text-align:right;"| 3
| style="text-align:right;"| [[7 (szám)|7]]
| style="text-align:right;"| 1
| i. e. 430 körül
| Ókori görög matematikusok
|
|-
| style="text-align:right;"| 3
| style="text-align:right;"| 5
| style="text-align:right;"| [[31 (szám)|31]]
| style="text-align:right;"| 2
| i. e. 300 körül
| Ókori görög matematikusok<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 (szám)|127]]
| style="text-align:right;"| 3
| i. e. 300 körül
| Ókori görög matematikusok<ref name="Euclid"/>
|
|-
| style="text-align:right;"| 5
| style="text-align:right;"| 13
| style="text-align:right;"| [[8191 (szám)|8191]]
| style="text-align:right;"| 4
| 1456
| ismeretlen
| [[Trial division]]
|-
| style="text-align:right;"| 6
| style="text-align:right;"| 17
| style="text-align:right;"| [[131071 (szám)|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#{{Halott link|url=http://fermi.imss.fi.it/rd/bdv?%2Fbdviewer%40selid=1373775 |date=2019-05 }}</ref>
| [[Pietro Cataldi]]
| Trial division<ref>pp. 13–18 in ''Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo'' 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#{{Halott link|url=http://fermi.imss.fi.it/rd/bdv?%2Fbdviewer%40selid=1373775 |date=2019-05 }}</ref>
|-
| style="text-align:right;"| 7
| style="text-align:right;"| 19
| style="text-align:right;"| [[524287 (szám)|524287]]
| style="text-align:right;"| 6
| 1588
| Pietro Cataldi
| Trial division<ref>pp. 18–22 in ''Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo'' 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#{{Halott link|url=http://fermi.imss.fi.it/rd/bdv?%2Fbdviewer%40selid=1373775 |date=2019-05 }}</ref>
|-
| style="text-align:right;"| 8
| style="text-align:right;"| 31
| style="text-align:right;"| [[2147483647 (szám)|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 {{Wayback|url=http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv%2F1772&seite%3Aint=36 |date=20120331144007 }} 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 Mihejevics Pervusin|Ivan Pervusin]]
| [[Lucas sequence]]s
|-
| style="text-align:right;"| 10
| style="text-align:right;"| 89
| style="text-align:right;"| 618970019642690137449562111
| style="text-align:right;"| 27
| 1911. június<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. június 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. január 10.<ref>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.</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. január 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 Robinson]]
| [[Lucas–Lehmer prímteszt|Lucas–Lehmer prímteszt (LLT)]] / [[SWAC]]
|-
| style="text-align:right;"| 14
| style="text-align:right;"| 607
| style="text-align:right;"| 531137992816...219031728127
| style="text-align:right;"| 183
| 1952. január 30.<ref name="autogenerated4" />
| Raphael Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 15
| style="text-align:right;"| {{szám|1279}}
| style="text-align:right;"| 104079321946...703168729087
| style="text-align:right;"| 386
| 1952. június 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 Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 16
| style="text-align:right;"| {{szám|2203}}
| style="text-align:right;"| 147597991521...686697771007
| style="text-align:right;"| 664
| 1952. október 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 Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 17
| style="text-align:right;"| {{szám|2281}}
| style="text-align:right;"| 446087557183...418132836351
| style="text-align:right;"| 687
| 1952. október 9.<ref name="autogenerated2" />
| Raphael Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 18
| style="text-align:right;"| {{szám|3217}}
| style="text-align:right;"| 259117086013...362909315071
| style="text-align:right;"| 969
| 1957. szeptember 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> &minus; 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;"| {{szám|4253}}
| style="text-align:right;"| 190797007524...815350484991
| style="text-align:right;"| {{szám|1281}}
| 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;"| {{szám|4423}}
| style="text-align:right;"| 285542542228...902608580607
| style="text-align:right;"| {{szám|1332}}
| 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;"| {{szám|9689}}
| style="text-align:right;"| 478220278805...826225754111
| style="text-align:right;"| {{szám|2917}}
| 1963. május 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 Gillies]]
| LLT / [[ILLIAC II]]
|-
| style="text-align:right;"| 22
| style="text-align:right;"| {{szám|9941}}
| style="text-align:right;"| 346088282490...883789463551
| style="text-align:right;"| 2,993
| 1963. május 16.<ref name="autogenerated3" />
| Donald Gillies
| LLT / ILLIAC II
|-
| style="text-align:right;"| 23
| style="text-align:right;"| {{szám|11213}}
| style="text-align:right;"| 281411201369...087696392191
| style="text-align:right;"| {{szám|3376}}
| 1963. június 2.<ref name="autogenerated3" />
| Donald Gillies
| LLT / ILLIAC II
|-
| style="text-align:right;"| 24
| style="text-align:right;"| {{szám|19937}}
| style="text-align:right;"| 431542479738...030968041471
| style="text-align:right;"| {{szám|6002}}
| 1971. március 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;"| {{szám|21701}}
| style="text-align:right;"| 448679166119...353511882751
| style="text-align:right;"| {{szám|6533}}
| 1978. október 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;"| {{szám|23209}}
| style="text-align:right;"| 402874115778...523779264511
| style="text-align:right;"| {{szám|6987}}
| 1979. február 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;"| {{szám|44497}}
| style="text-align:right;"| 854509824303...961011228671
| style="text-align:right;"| {{szám|13395}}
| 1979. április 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 Lewis Nelson ]] & [[David Slowinski]]
| LLT / [[Cray 1]]
|-
| style="text-align:right;"| 28
| style="text-align:right;"| {{szám|86243}}
| style="text-align:right;"| 536927995502...709433438207
| style="text-align:right;"| {{szám|25962}}
| 1982. szeptember 25.
| David Slowinski
| LLT / Cray 1
|-
| style="text-align:right;"| 29
| style="text-align:right;"| {{szám|110503}}
| style="text-align:right;"| 521928313341...083465515007
| style="text-align:right;"| {{szám|33265}}
| 1988. január 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;"| {{szám|39751}}
| 1983. szeptember 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;"| {{szám|216091}}
| style="text-align:right;"| 746093103064...103815528447
| style="text-align:right;"| {{szám|65050}}
| 1985. szeptember 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> &minus; 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;"| {{szám|756839}}
| style="text-align:right;"| 174135906820...328544677887
| style="text-align:right;"| {{szám|227832}}
| 1992. február 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;"| {{szám|859433}}
| style="text-align:right;"| 129498125604...243500142591
| style="text-align:right;"| {{szám|258716}}
| 1994. január 4.<ref>Chris Caldwell, [http://www.math.unicaen.fr/~reyssat/largest.html The Largest Known Primes] {{Wayback|url=http://www.math.unicaen.fr/~reyssat/largest.html |date=19981202090207 }}.</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;"| {{szám|1257787}}
| style="text-align:right;"| 412245773621...976089366527
| style="text-align:right;"| {{szám|378632}}
| 1996. szeptember 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;"| {{szám|1398269}}
| style="text-align:right;"| 814717564412...868451315711
| style="text-align:right;"| {{szám|420921}}
| 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;"| {{szám|2976221}}
| style="text-align:right;"| 623340076248...743729201151
| style="text-align:right;"| {{szám|895932}}
| 1997. augusztus 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;"| {{szám|3021377}}
| style="text-align:right;"| 127411683030...973024694271
| style="text-align:right;"| {{szám|909526}}
| 1998. január 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;"| {{szám|6972593}}
| style="text-align:right;"| 437075744127...142924193791
| style="text-align:right;"| {{szám|2098960}}
| 1999. június 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;"| {{szám|13466917}}
| style="text-align:right;"| 924947738006...470256259071
| style="text-align:right;"| {{szám|4053946}}
| 2001. november 14.
| GIMPS / [[Michael Cameron (matematikus)|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;"| {{szám|20996011}}
| style="text-align:right;"| 125976895450...762855682047
| style="text-align:right;"| {{szám|6320430}}
| 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;"| {{szám|24036583}}
| style="text-align:right;"| 299410429404...882733969407
| style="text-align:right;"| {{szám|7235733}}
| 2004. május 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;"| {{szám|25964951}}
| style="text-align:right;"| 122164630061...280577077247
| style="text-align:right;"| {{szám|7816230}}
| 2005. február 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;"| {{szám|30402457}}
| style="text-align:right;"| 315416475618...411652943871
| style="text-align:right;"| {{szám|9152052}}
| 2005. december 15.
| GIMPS / [[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;"| {{szám|32582657}}
| style="text-align:right;"| 124575026015...154053967871
| style="text-align:right;"| {{szám|9808358}}
| 2006. szeptember 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;"| {{szám|37156667}}
| style="text-align:right;"| 202254406890...022308220927
| style="text-align:right;"| {{szám|11185272}}
| 2008. szeptember 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">Nincs igazolva, hogy nem létezik más Mersenne-prím a 45. ({{math|''M''<sub>{{szám|37156667}}</sub>}}) és az 51. ({{math|''M''<sub>{{szám|82589933}}</sub>}}) sorszámú között; a sorrend tehát nem végleges.</ref>
| style="text-align:right;"| {{szám|42643801}}
| style="text-align:right;"| 169873516452...765562314751
| style="text-align:right;"| {{szám|12837064}}
| 2009. április 12.<ref group="n" name="date_issue">Az {{math|''M''<sub>{{szám|42643801}}</sub>}} prímet 2009. április 12-én találta meg egy számítógép, de ezt a tényt június 4-ig nem vette észre senki, így mindkét dátum tekinthető a „felfedezés” időpontjának.</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 Stig M. Valstad néven is ismert.</ref>
| LLT / Prime95 on 3&nbsp;GHz Core 2
|-
| style="text-align:right;"| 47<ref group="n" name="unverified_index" />
| style="text-align:right;"| {{szám|43112609}}
| style="text-align:right;"| 316470269330...166697152511
| style="text-align:right;"| {{szám|12978189}}
| 2008. augusztus 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;"| {{szám|57885161}}
| style="text-align:right;"| 581887266232...071724285951
| style="text-align:right;"| {{szám|17425170}}
| 2013. január 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> &minus; 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;"| {{szám|74207281}}
| style="text-align:right;"| 300376418084...391086436351
| style="text-align:right;"| {{szám|22338618}}
| 2015. szeptember 17.<ref group="n" name="date_issue_2>Az {{math|''M''<sub>{{szám|74207281}}</sub>}} prímet 2015. szeptember 17-én találta meg egy számítógép, de ezt a tényt 2016. január 7-ig nem vette észre senki, így mindkét dátum tekinthető a „felfedezés” időpontjának. A GIMPS a későbbit tekinti hivatalosnak.</ref>
| GIMPS / Curtis Cooper<ref name="GIMPS-2016">{{cite web |last=Cooper |first=Curtis |title=Mersenne Prime Number discovery – 2<sup>74207281</sup>&nbsp;−&nbsp;1 is Prime! |url=http://www.mersenne.org/primes/?press=M74207281 |date=7 January 2016 |work=Mersenne Research, Inc. |accessdate=22 January 2016 }}</ref>
| LLT / Prime95 on Intel [[Core i7-4790]]
|-
| style="text-align:right;"| 50<ref group="n" name="unverified_index" />
| style="text-align:right;"| {{szám|77232917}}
| style="text-align:right;"| 467333183359...069762179071
| style="text-align:right;"| {{szám|23249425}}
| 2017. december 26.
| GIMPS / [[Jon Pace]]<ref name="GIMPS-2017">{{cite web |title=GIMPS Project Discovers Largest Known Prime Number: 2<sup>77,232,917</sup>-1 |url=https://www.mersenne.org/primes/press/M77232917.html |date=3 January 2018 |work=Mersenne Research, Inc. |accessdate=3 January 2018 }}</ref>
| LLT / Prime95 on 3.3 GHz Intel [[Core i5-6600]]<ref>{{cite web|url=https://www.mersenne.org/primes/|title=List of known Mersenne prime numbers|accessdate=3 January 2018}}</ref>
|-
| style="text-align:right;"| 51<ref group="n" name="unverified_index" />
| style="text-align:right;"| {{szám|82589933}}
| style="text-align:right;"| 148894445742...325217902591
| style="text-align:right;"| {{szám|24862048}}
| 2018. december 7.
| GIMPS / Patrick Laroche<ref name="GIMPS-2018">{{cite web |title=GIMPS Project Discovers Largest Known Prime Number: 2<sup>82,589,933</sup>-1 |url=https://www.mersenne.org/primes/press/M82589933.html |date=21 December 2018 |work=Mersenne Research, Inc. |accessdate=21 December 2018 }}</ref>
| LLT / Prime95 on Intel Core i5-4590T
|-
| style="text-align:right;"| 52<ref group="n" name="unverified_index" />
| style="text-align:right;"| {{szám|136279841}}
| style="text-align:right;"| 
| style="text-align:right;"| {{szám|41024320}}
| 2024. október 11.
|GIMPS / Luke Durant<ref>{{CitWeb|tit=Csak leírni több hónapba telne: megvan az eddigi legnagyobb prímszám, 41 millió számjegyből áll|url=https://hvg.hu/tudomany/20241025_eddigi-legnagyobb-primszam-mersenne-primek-gimps-luke-durant|accd=2024-10-25|date=2024-10-25}}</ref>
|
|}

<references group="n" />

== Jegyzetek ==
{{jegyzetek}}

== Források ==
*http://www.mersenne.org/

{{Portál|Matematika}}
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[[Kategória:Prímszámok]]