Mersenne-prímek
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A matematika megoldatlan problémája: Létezik-e végtelen sok Mersenne-prím? (A matematika további megoldatlan problémái)
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A matematikában Mersenne-prímeknek nevezzük a kettő-hatványnál eggyel kisebb, azaz a 2n ‒ 1 alakban felírható prímszámokat, ahol n szintén prímszám. A nevüket Marin Mersenne (1588–1648) francia szerzetes, matematikus, fizikus után kapták.
Matematikai alapok
[szerkesztés]Például a 31 (prímszám) = 32 ‒ 1 = 25 ‒ 1, és 5 szintén prím, ezért a 31 egy Mersenne-prím; hasonlóan, 7 = 8 ‒ 1 = 23 ‒ 1. Másrészt 2047 = 2048 ‒ 1 = 211 ‒ 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.[1]
A Mersenne-prím definíciójában a kikötés, hogy n szükségképpen prím, elhagyható, ugyanis minden összetett n esetén elemi módon felbontható:
Általánosabban, a Mersenne-számok (nem feltétlenül prímek, de lehetnek azok is) olyan természetes számok, amelyek eggyel kisebbek egy kettő-hatványnál, tehát Mn = 2n − 1. (A legtöbb forrás a Mersenne-számoknál is megköveteli, hogy az n prímszám legyen.)
A Mersenne-prímek listája
[szerkesztés]![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Primes.png/450px-Primes.png)
Sorszám | Hatványkitevő (p) | Mersenne-prím (Mp) | Számjegy (Mp) | Felfedezés éve | Felfedező | Használt módszer |
---|---|---|---|---|---|---|
1 | 2 | 3 | 1 | i. e. 430 körül | Ókori görög matematikusok | |
2 | 3 | 7 | 1 | i. e. 430 körül | Ókori görög matematikusok | |
3 | 5 | 31 | 2 | i. e. 300 körül | Ókori görög matematikusok[2] | |
4 | 7 | 127 | 3 | i. e. 300 körül | Ókori görög matematikusok[2] | |
5 | 13 | 8191 | 4 | 1456 | ismeretlen | Trial division |
6 | 17 | 131071 | 6 | 1588[3] | Pietro Cataldi | Trial division[4] |
7 | 19 | 524287 | 6 | 1588 | Pietro Cataldi | Trial division[5] |
8 | 31 | 2147483647 | 10 | 1772 | Leonhard Euler[6][7] | Enhanced trial division[8] |
9 | 61 | 2305843009213693951 | 19 | 1883. november[9] | Ivan Pervusin | Lucas sequences |
10 | 89 | 618970019642690137449562111 | 27 | 1911. június[10] | Ralph Ernest Powers | Lucas sequences |
11 | 107 | 162259276829...578010288127 | 33 | 1914. június 1.[11][12][13] | Ralph Ernest Powers[14] | Lucas sequences |
12 | 127 | 170141183460...715884105727 | 39 | 1876. január 10.[15] | Édouard Lucas | Lucas sequences |
13 | 521 | 686479766013...291115057151 | 157 | 1952. január 30.[16] | Raphael Robinson | Lucas–Lehmer prímteszt (LLT) / SWAC |
14 | 607 | 531137992816...219031728127 | 183 | 1952. január 30.[16] | Raphael Robinson | LLT / SWAC |
15 | 1279 | 104079321946...703168729087 | 386 | 1952. június 25.[17] | Raphael Robinson | LLT / SWAC |
16 | 2203 | 147597991521...686697771007 | 664 | 1952. október 7.[18] | Raphael Robinson | LLT / SWAC |
17 | 2281 | 446087557183...418132836351 | 687 | 1952. október 9.[18] | Raphael Robinson | LLT / SWAC |
18 | 3217 | 259117086013...362909315071 | 969 | 1957. szeptember 8.[19] | Hans Riesel | LLT / BESK |
19 | 4253 | 190797007524...815350484991 | 1281 | 1961. november 3.[20][21] | Alexander Hurwitz | LLT / IBM 7090 |
20 | 4423 | 285542542228...902608580607 | 1332 | 1961. november 3.[20][21] | Alexander Hurwitz | LLT / IBM 7090 |
21 | 9689 | 478220278805...826225754111 | 2917 | 1963. május 11.[22] | Donald Gillies | LLT / ILLIAC II |
22 | 9941 | 346088282490...883789463551 | 2,993 | 1963. május 16.[22] | Donald Gillies | LLT / ILLIAC II |
23 | 11 213 | 281411201369...087696392191 | 3376 | 1963. június 2.[22] | Donald Gillies | LLT / ILLIAC II |
24 | 19 937 | 431542479738...030968041471 | 6002 | 1971. március 4.[23] | Bryant Tuckerman | LLT / IBM 360/91 |
25 | 21 701 | 448679166119...353511882751 | 6533 | 1978. október 30.[24] | Landon Curt Noll & Laura Nickel | LLT / CDC Cyber 174 |
26 | 23 209 | 402874115778...523779264511 | 6987 | 1979. február 9.[25] | Landon Curt Noll | LLT / CDC Cyber 174 |
27 | 44 497 | 854509824303...961011228671 | 13 395 | 1979. április 8.[26][27] | Harry Lewis Nelson & David Slowinski | LLT / Cray 1 |
28 | 86 243 | 536927995502...709433438207 | 25 962 | 1982. szeptember 25. | David Slowinski | LLT / Cray 1 |
29 | 110 503 | 521928313341...083465515007 | 33 265 | 1988. január 29.[28][29] | Walter Colquitt & Luke Welsh | LLT / NEC SX-2[30] |
30 | 132,049 | 512740276269...455730061311 | 39 751 | 1983. szeptember 19.[31] | David Slowinski | LLT / Cray X-MP |
31 | 216 091 | 746093103064...103815528447 | 65 050 | 1985. szeptember 1.[32][33] | David Slowinski | LLT / Cray X-MP/24 |
32 | 756 839 | 174135906820...328544677887 | 227 832 | 1992. február 17. | David Slowinski & Paul Gage | LLT / Harwell Lab's Cray-2[34] |
33 | 859 433 | 129498125604...243500142591 | 258 716 | 1994. január 4.[35][36][37] | David Slowinski & Paul Gage | LLT / Cray C90 |
34 | 1 257 787 | 412245773621...976089366527 | 378 632 | 1996. szeptember 3.[38] | David Slowinski & Paul Gage[39] | LLT / Cray T94 |
35 | 1 398 269 | 814717564412...868451315711 | 420 921 | 1996. november 13. | GIMPS / Joel Armengaud[40] | LLT / Prime95 on 90 MHz Pentium |
36 | 2 976 221 | 623340076248...743729201151 | 895 932 | 1997. augusztus 24. | GIMPS / Gordon Spence[41] | LLT / Prime95 on 100 MHz Pentium |
37 | 3 021 377 | 127411683030...973024694271 | 909 526 | 1998. január 27. | GIMPS / Roland Clarkson[42] | LLT / Prime95 on 200 MHz Pentium |
38 | 6 972 593 | 437075744127...142924193791 | 2 098 960 | 1999. június 1. | GIMPS / Nayan Hajratwala[43] | LLT / Prime95 on 350 MHz Pentium II IBM Aptiva |
39 | 13 466 917 | 924947738006...470256259071 | 4 053 946 | 2001. november 14. | GIMPS / Michael Cameron[44] | LLT / Prime95 on 800 MHz Athlon T-Bird |
40 | 20 996 011 | 125976895450...762855682047 | 6 320 430 | 2003. november 17. | GIMPS / Michael Shafer[45] | LLT / Prime95 on 2 GHz Dell Dimension |
41 | 24 036 583 | 299410429404...882733969407 | 7 235 733 | 2004. május 15. | GIMPS / Josh Findley[46] | LLT / Prime95 on 2.4 GHz Pentium 4 |
42 | 25 964 951 | 122164630061...280577077247 | 7 816 230 | 2005. február 18. | GIMPS / Martin Nowak[47] | LLT / Prime95 on 2.4 GHz Pentium 4 |
43 | 30 402 457 | 315416475618...411652943871 | 9 152 052 | 2005. december 15. | GIMPS / Curtis Cooper & Steven Boone[48] | LLT / Prime95 on 2 GHz Pentium 4 |
44 | 32 582 657 | 124575026015...154053967871 | 9 808 358 | 2006. szeptember 4. | GIMPS / Curtis Cooper & Steven Boone[49] | LLT / Prime95 on 3 GHz Pentium 4 |
45 | 37 156 667 | 202254406890...022308220927 | 11 185 272 | 2008. szeptember 6. | GIMPS / Hans-Michael Elvenich[50] | LLT / Prime95 on 2.83 GHz Core 2 Duo |
46[n 1] | 42 643 801 | 169873516452...765562314751 | 12 837 064 | 2009. április 12.[n 2] | GIMPS / Odd M. Strindmo[51][n 3] | LLT / Prime95 on 3 GHz Core 2 |
47[n 1] | 43 112 609 | 316470269330...166697152511 | 12 978 189 | 2008. augusztus 23. | GIMPS / Edson Smith[50] | LLT / Prime95 on Dell Optiplex 745 |
48[n 1] | 57 885 161 | 581887266232...071724285951 | 17 425 170 | 2013. január 25. | GIMPS / Curtis Cooper[52] | LLT / Prime95 on 3 GHz Intel Core2 Duo E8400[53] |
49[n 1] | 74 207 281 | 300376418084...391086436351 | 22 338 618 | 2015. szeptember 17.[n 4] | GIMPS / Curtis Cooper[54] | LLT / Prime95 on Intel Core i7-4790 |
50[n 1] | 77 232 917 | 467333183359...069762179071 | 23 249 425 | 2017. december 26. | GIMPS / Jon Pace[55] | LLT / Prime95 on 3.3 GHz Intel Core i5-6600[56] |
51[n 1] | 82 589 933 | 148894445742...325217902591 | 24 862 048 | 2018. december 7. | GIMPS / Patrick Laroche[57] | LLT / Prime95 on Intel Core i5-4590T |
- ↑ a b c d e f Nincs igazolva, hogy nem létezik más Mersenne-prím a 45. (M37 156 667) és az 51. (M82 589 933) sorszámú között; a sorrend tehát nem végleges.
- ↑ Az M42 643 801 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.
- ↑ Strindmo Stig M. Valstad néven is ismert.
- ↑ Az M74 207 281 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.
Jegyzetek
[szerkesztés]- ↑ The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992; see Caldwell, "The Largest Known Prime by Year: A Brief History" from the Prime Pages website, University of Tennessee at Martin.
- ↑ a b Euclid's Elements, Book IX, Proposition 36
- ↑ "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]
- ↑ 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]
- ↑ 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]
- ↑ http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 Archiválva 2012. március 31-i dátummal a Wayback Machine-ben 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.
- ↑ Chris K. Caldwell: Modular restrictions on Mersenne divisors. Primes.utm.edu. (Hozzáférés: 2011. május 21.)
- ↑ “En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre 261 − 1 = 2305843009213693951 est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. http://www.biodiversitylibrary.org/item/107789#page/277/mode/1up [retrieved 2012-09-17] See also Mélanges mathématiques et astronomiques tirés du Bulletin de l’Académie impériale des sciences de St.-Pétersbourg v. 6 (1881–1888), pp. 553–554. See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48
- ↑ Powers, R. E. (1911. január 1.). „The Tenth Perfect Number”. The American Mathematical Monthly 18 (11), 195–197. o. DOI:10.2307/2972574.
- ↑ "M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1er Juin un cablógramme à M. Bromwich [secretary of London Mathematical Society] pour M107. Sur ma demande, ces deux auteurs m’ont adressé leurs remarquables résultats, et je m’empresse de les publier dans nos colonnes, avec nos felicitations." p. 103, André Gérardin, Nombres de Mersenne pp. 85, 103–108 in Sphinx-Œdipe. [Journal mensuel de la curiosité, de concours & de mathématiques.] v. 9, No. 1, 1914.
- ↑ "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.
- ↑ The Prime Pages, M107: Fauquembergue or Powers?.
- ↑ 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.
- ↑ a b "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]
- ↑ "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]
- ↑ a b "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]
- ↑ "On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number M3217 = 23217 − 1 is a prime." Hans Riesel, A New Mersenne Prime, Mathematics of Computation, vol. 12 (1958), p. 60, http://www.ams.org/journals/mcom/1958-12-061/S0025-5718-1958-0099752-6/S0025-5718-1958-0099752-6.pdf [Retrieved 2012-09-18]
- ↑ a b 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.
- ↑ a b "If p is prime, Mp = 2p − 1 is called a Mersenne number. The primes M4253 and M4423 were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, New Mersenne Primes, Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18]
- ↑ a b c "The primes M9689, M9941, and M11213 which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois." Donald B. Gillies, Three New Mersenne Primes and a Statistical Theory, Mathematics of Computation, vol. 18, No. 85 (1964), pp. 93–97, http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0159774-6/S0025-5718-1964-0159774-6.pdf [Retrieved 2012-09-18]
- ↑ "On the evening of March 4, 1971, a zero Lucas-Lehmer residue for p = p24 = 19937 was found. Hence, M19937 is the 24th Mersenne prime." Bryant Tuckerman, The 24th Mersenne Prime, Proceedings of the National Academy of Sciences of the United States of America, vol. 68:10 (1971), pp. 2319–2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]
- ↑ "On October 30, 1978 at 9:40 pm, we found M21701 to be prime. The CPU time required for this test was 7:40:20. Tuckerman and Lehmer later provided confirmation of this result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
- ↑ "Of the 125 remaining Mp only M23209 was found to be prime. The test was completed on February 9, 1979 at 4:06 after 8:39:37 of CPU time. Lehmer and McGrogan later confirmed the result." Curt Noll and Laura Nickel, The 25th and 26th Mersenne Primes, Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]
- ↑ David 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.
- ↑ "An FFT containing 8192 complex elements, which was the minimum size required to test M110503, ran approximately 11 minutes on the SX-2. The discovery of M110503 (January 29, 1988) has been confirmed." W. N. Colquitt and L. Welsh, Jr., A New Mersenne Prime, Mathematics of Computation, vol. 56, No. 194 (April 1991), pp. 867–870, http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068823-9/S0025-5718-1991-1068823-9.pdf [Retrieved 2012-09-18]
- ↑ "This week, two computer experts found the 31st Mersenne prime. But to their surprise, the newly discovered prime number falls between two previously known Mersenne primes. It occurs when p = 110,503, making it the third-largest Mersenne prime known." I. Peterson, Priming for a lucky strike Science News; 2/6/88, Vol. 133 Issue 6, pp. 85–85. http://ehis.ebscohost.com/ehost/detail?vid=3&hid=23&sid=9a9d7493-ffed-410b-9b59-b86c63a93bc4%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8824187 [Retrieved 2012-09-18]
- ↑ Mersenne Prime Numbers. Omes.uni-bielefeld.de, 2011. január 5. (Hozzáférés: 2011. május 21.)
- ↑ "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]
- ↑ "The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form 2p − 1, where the exponent p is also a prime number. For instance, 127 is a Mersenne number for which the exponent is 7. The record prime number's exponent is 216,091." I. Peterson, Prime time for supercomputers Science News; 9/28/85, Vol. 128 Issue 13, p. 199. http://ehis.ebscohost.com/ehost/detail?vid=4&hid=22&sid=c11090a2-4670-469f-8f75-947b593a56a0%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8840537 [Retrieved 2012-09-18]
- ↑ "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]
- ↑ The Prime Pages, The finding of the 32nd Mersenne.
- ↑ Chris Caldwell, The Largest Known Primes Archiválva 1998. december 2-i dátummal a Wayback Machine-ben.
- ↑ Crays press release
- ↑ Slowinskis email
- ↑ Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
- ↑ The Prime Pages, A Prime of Record Size! 21257787 – 1.
- ↑ GIMPS Discovers 35th Mersenne Prime.
- ↑ GIMPS Discovers 36th Known Mersenne Prime.
- ↑ GIMPS Discovers 37th Known Mersenne Prime.
- ↑ GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
- ↑ GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
- ↑ GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583 – 1.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951 – 1.
- ↑ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457 – 1.
- ↑ GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657 – 1.
- ↑ a b Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
- ↑ "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]
- ↑ GIMPS Discovers 48th Mersenne Prime, 257,885,161 − 1 is now the Largest Known Prime.. Great Internet Mersenne Prime Search. (Hozzáférés: 2016. január 19.)
- ↑ List of known Mersenne prime numbers. (Hozzáférés: 2014. november 29.)
- ↑ Cooper, Curtis: Mersenne Prime Number discovery – 274207281 − 1 is Prime!. Mersenne Research, Inc., 2016. január 7. (Hozzáférés: 2016. január 22.)
- ↑ GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1. Mersenne Research, Inc., 2018. január 3. (Hozzáférés: 2018. január 3.)
- ↑ List of known Mersenne prime numbers. (Hozzáférés: 2018. január 3.)
- ↑ GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1. Mersenne Research, Inc., 2018. december 21. (Hozzáférés: 2018. december 21.)