Mersenne-prímek

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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). A modern kori matematikában a legnagyobb ismert prímszám gyakran Mersenne-prím volt.

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]

Ábra az adott évben ismert legnagyobb Mersenne-prím számjegyeinek számáról
Sorszám Hatványkitevő (p) Mersenne-prím (Mp) Számjegy (Mp) Felfedezés éve Felfedező Használt módszer
1 2 3 1 ie. 430 körül Ókori görög matematikusok
2 3 7 1 ie. 430 körül Ókori görög matematikusok
3 5 31 2 ie. 300 körül Ókori görög matematikusok[1]
4 7 127 3 ie. 300 körül Ókori görög matematikusok[1]
5 13 8191 4 1456 ismeretlen Trial division
6 17 131071 6 1588[2] Pietro Cataldi Trial division[3]
7 19 524287 6 1588 Pietro Cataldi Trial division[4]
8 31 2147483647 10 1772 Leonhard Euler[5][6] Enhanced trial division[7]
9 61 2305843009213693951 19 1883. november[8] Ivan Pervusin Lucas sequences
10 89 618970019642...137449562111 27 1911. június[9] Ralph Ernest Powers Lucas sequences
11 107 162259276829...578010288127 33 1914. június 1.[10][11][12] Ralph Ernest Powers[13] Lucas sequences
12 127 170141183460...715884105727 39 1876. január 10.[14] Édouard Lucas Lucas sequences
13 521 686479766013...291115057151 157 1952. január 30.[15] Raphael Robinson Lucas–Lehmer prímteszt (LLT) / SWAC
14 607 531137992816...219031728127 183 1952. január 30.[15] Raphael Robinson LLT / SWAC
15 1279 104079321946...703168729087 386 1952. június 25.[16] Raphael Robinson LLT / SWAC
16 2203 147597991521...686697771007 664 1952. október 7.[17] Raphael Robinson LLT / SWAC
17 2281 446087557183...418132836351 687 1952. október 9.[17] Raphael Robinson LLT / SWAC
18 3217 259117086013...362909315071 969 1957. szeptember 8.[18] Hans Riesel LLT / BESK
19 4253 190797007524...815350484991 1281 1961. november 3.[19][20] Alexander Hurwitz LLT / IBM 7090
20 4423 285542542228...902608580607 1332 1961. november 3.[19][20] Alexander Hurwitz LLT / IBM 7090
21 9689 478220278805...826225754111 2917 1963. május 11.[21] Donald Gillies LLT / ILLIAC II
22 9941 346088282490...883789463551 2,993 1963. május 16.[21] Donald Gillies LLT / ILLIAC II
23 11 213 281411201369...087696392191 3376 1963. június 2.[21] Donald Gillies LLT / ILLIAC II
24 19 937 431542479738...030968041471 6002 1971. március 4.[22] Bryant Tuckerman LLT / IBM 360/91
25 21 701 448679166119...353511882751 6533 1978. október 30.[23] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174
26 23 209 402874115778...523779264511 6987 1979. február 9.[24] Landon Curt Noll LLT / CDC Cyber 174
27 44 497 854509824303...961011228671 13 395 1979. április 8.[25][26] 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.[27][28] Walter Colquitt & Luke Welsh LLT / NEC SX-2[29]
30 132,049 512740276269...455730061311 39 751 1983. szeptember 19.[30] David Slowinski LLT / Cray X-MP
31 216 091 746093103064...103815528447 65 050 1985. szeptember 1.[31][32] David Slowinski LLT / Cray X-MP/24
32 756 839 174135906820...328544677887 227 832 1992. feruár 17. David Slowinski & Paul Gage LLT / Harwell Lab's Cray-2[33]
33 859 433 129498125604...243500142591 258 716 1994. január 4.[34][35][36] David Slowinski & Paul Gage LLT / Cray C90
34 1 257 787 412245773621...976089366527 378 632 1996. szeptember 3.[37] David Slowinski & Paul Gage[38] LLT / Cray T94
35 1 398 269 814717564412...868451315711 420 921 1996. november 13. GIMPS / Joel Armengaud[39] LLT / Prime95 on 90 MHz Pentium
36 2 976 221 623340076248...743729201151 895 932 1997. augusztus 24. GIMPS / Gordon Spence[40] LLT / Prime95 on 100 MHz Pentium
37 3 021 377 127411683030...973024694271 909 526 1998. január 27. GIMPS / Roland Clarkson[41] LLT / Prime95 on 200 MHz Pentium
38 6 972 593 437075744127...142924193791 2 098 960 1999. június 1. GIMPS / Nayan Hajratwala[42] LLT / Prime95 on 350 MHz Pentium II IBM Aptiva
39 13 466 917 924947738006...470256259071 4 053 946 2001. november 14. GIMPS / Michael Cameron[43] LLT / Prime95 on 800 MHz Athlon T-Bird
40 20 996 011 125976895450...762855682047 6 320 430 2003. november 17. GIMPS / Michael Shafer[44] LLT / Prime95 on 2 GHz Dell Dimension
41 24 036 583 299410429404...882733969407 7 235 733 2004. május 15. GIMPS / Josh Findley[45] LLT / Prime95 on 2.4 GHz Pentium 4
42 25 964 951 122164630061...280577077247 7 816 230 2005. február 18. GIMPS / Martin Nowak[46] 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[47] LLT / Prime95 on 2 GHz Pentium 4
44 32 582 657 124575026015...154053967871 9 808 358 2006. szeptember 4. GIMPS / Curtis Cooper & Steven Boone[48] LLT / Prime95 on 3 GHz Pentium 4
45 37 156 667 202254406890...022308220927 11 185 272 2008. szeptember 6. GIMPS / Hans-Michael Elvenich[49] 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[50][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[49] LLT / Prime95 on Dell Optiplex 745
48[n 1] 57 885 161 581887266232...071724285951 17 425 170 2013. january 25. GIMPS / Curtis Cooper[51] LLT / Prime95 on 3 GHz Intel Core2 Duo E8400[52]
49[n 1] 74 207 281 300376418084...391086436351 22 338 618 2015. szeptember 17.[n 4] GIMPS / Curtis Cooper[53] LLT / Prime95 on Intel Core i7-4790
50[n 1] 77 232 917 467333183359...069762179071 23 249 425 2017. december 26. GIMPS / Jon Pace[54] LLT / Prime95 on 3.3 GHz Intel Core i5-6600[55]
  1. ^ a b c d e It is not verified whether any undiscovered Mersenne primes exist between the 45th (M37,156,667) and the 50th (M77,232,917) on this chart; the ranking is therefore provisional.
  2. M42,643,801 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.
  3. Strindmo also uses the alias Stig M. Valstad.
  4. M74,207,281 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.

Jegyzetek[szerkesztés]

  1. ^ a b Euclid's Elements, Book IX, Proposition 36
  2. "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#
  3. pp. 13–18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  4. pp. 18–22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  5. http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1772&seite:int=36 Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres 1772, pp. 35–36 EULER, Leonhard: Extrait d'une lettre à M. Bernoulli, concernant le Mémoire imprimé parmi ceux de 1771. p. 318 [intitulé: Recherches sur les diviseurs de quelques nombres très grands compris dans la somme de la progression géométrique 1 + 101 + 102 + 103 + ... + 10T = S]. Retrieved 2011-10-02.
  6. http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.
  7. Chris K. Caldwell: Modular restrictions on Mersenne divisors. Primes.utm.edu. (Hozzáférés: 2011. május 21.)
  8. “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
  9. Powers, R. E. (1911. január 1.). „The Tenth Perfect Number”. The American Mathematical Monthly 18 (11), 195–197. o. DOI:10.2307/2972574.  
  10. "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.
  11. "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]
  12. 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.
  13. The Prime Pages, M107: Fauquembergue or Powers?.
  14. 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.
  15. ^ 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]
  16. "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]
  17. ^ 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]
  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]
  19. ^ 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.
  20. ^ 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]
  21. ^ 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]
  22. "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]
  23. "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]
  24. "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]
  25. David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978–79, pp. 258–261, MR 80g #10013
  26. "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.
  27. "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]
  28. "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]
  29. Mersenne Prime Numbers. Omes.uni-bielefeld.de, 2011. január 5. (Hozzáférés: 2011. május 21.)
  30. "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]
  31. "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]
  32. "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]
  33. The Prime Pages, The finding of the 32nd Mersenne.
  34. Chris Caldwell, The Largest Known Primes Archiválva 1998. december 2-i dátummal a Wayback Machine-ben.
  35. Crays press release
  36. Slowinskis email
  37. Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
  38. The Prime Pages, A Prime of Record Size! 21257787 – 1.
  39. GIMPS Discovers 35th Mersenne Prime.
  40. GIMPS Discovers 36th Known Mersenne Prime.
  41. GIMPS Discovers 37th Known Mersenne Prime.
  42. GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
  43. GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
  44. GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
  45. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583 – 1.
  46. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951 – 1.
  47. GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457 – 1.
  48. GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657 – 1.
  49. ^ a b Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
  50. "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]
  51. GIMPS Discovers 48th Mersenne Prime, 257,885,161 − 1 is now the Largest Known Prime.. Great Internet Mersenne Prime Search. (Hozzáférés: 2016. január 19.)
  52. List of known Mersenne prime numbers. (Hozzáférés: 2014. november 29.)
  53. Cooper, Curtis: Mersenne Prime Number discovery – 274207281 − 1 is Prime!. Mersenne Research, Inc., 2016. január 7. (Hozzáférés: 2016. január 22.)
  54. GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1. Mersenne Research, Inc., 2018. január 3. (Hozzáférés: 2018. január 3.)
  55. List of known Mersenne prime numbers. (Hozzáférés: 2018. január 3.)

Források[szerkesztés]