عدد مرسن الأولي

عدد مرسن الأولي
سُمي على اسممارين مرسن
تاريخ النشر1536[1]
مؤلف التطبيقرجيوس، هـ.
عدد المصطلحات المعروفة48
Conjectured number of termsInfinite
Subsequence ofأعداد مرسن
أول المصطلحات3، 7، 31، 127
أكبر مصطلح معروف257885161 − 1
OEIS indexA000668

في الرياضيات، عدد مرسين Mersenne number، هو عدد صحيح موجب أصغر من قوة العدد اثنين بواحد:

سميت هذه الأعداد نسبة لمارين مرسين وهو راهب فرنسي بدأ دراستها في بداية القرن السابع عشر.

بعض التعريفات لأعداد مرسين تشترط في الأس p أن يكون أوليا، بما أنه إذا كان p عددا مؤلفا فإن العدد يكون مؤلفا أيضا.

من المعلوم أنه إذا كان عددا أولياً فإن p هو عدد أولي أيضاً. بحلول أكتوبر عام 2009، اكتشف سبعة وأربعون عددا أولياً لمرسين فقط. أكبر عدد أولي معروف (ويساوي ) هو عدد أولي لمرسين. كل أعداد مرسين الأولية المكتشفة بعد 1997، اكتشفت بفضل مشروع البحث الكبير على الإنترنت عن أعداد مرسن الأولية.

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خلفية

الأعداد المثالية

تكمن أهمية أعداد ميرسين الأولية في ارتباطها بالأعداد المثالية. في القرن الرابع قبل الميلاد، برهن اقليدس على أنه إذا كان Mp عددا أوليا لميرسن، فإن

هو عدد مثالي زوجي. في القرن العاشر، يبدو أن ابن الهيثم كان أول من حاول تصنيف الأعداد المثالية الزوجية على شكل ( حيث هو عدد أولي. في القرن الثامن عشر، برهن ليونهارد أويلر على عكس هاته المبرهنة والذي ينص على أن كل عدد مثالي زوجي له هذا الشكل.


التاريخ

اعتقد عدد من الرياضيين السابقين أن العدد من الصورة يكون أوليا كلما كان n عددا أوليا، و لكن في 1536 أثبت ريجيوس ( Regius ) أن العدد : 2047 = 23.89 = ليس أوليا حيث أنه حاصل ضرب 23 و89، و في عام 1603 تحقق كاتالدي أن العددين و أوليان ، و استنتج كاتالدي و بشكل خاطئ أن العدد يكون أوليا لكل : n = 23,29,31,37 ، حيث أثبت فيرما في 1645 أن كاتالدي كان خاطئا بالنسبة للعددين n = 23,37 ، و أثبت أويلر في 1738 أن كاتالدي كان أيضا خاطئا بالنسبة للعدد n = 29 ، و في وقت لاحق أثبت أويلر أن كاتالدي كان مصيبا بالنسبة للعدد n = 31.

بمجيء الفرنسي مارين ميرسين (1588-1648)، حيث وضع في مقدمة أحد كتبه أن العدد يكون أوليا عندما : n = 2,3,5,7,13,17,19,31,67,127,257 ، و أنه مركب لكل الأعداد n < 257 الصحيحة، و رغم أن هذا التخمين من ميرسين كان خاطئا إلا أن اسمه ظل ملتصقا بهذه الأعداد حيث سميت باسمه.

كان واضحا أنه ليس بإمكان ميرسين التحقق من كل هذه الأعداد (n < 257) لصعوبة ذلك في عصر ميرسين. كذلك لم يكن بمقدور معاصريه التحقق من موضوعته، فبقيت كذلك إلى مائة سنة و ذلك عندما تحقق أويلر في 1750 من أن العدد التالي في قائمة ميرسين هو ، و بعد قرن آخر و في 1876 بين إدوارد لوكاس أن العدد أولي، و بعد سبع سنوات أثبت عالم الرياضيات الروسي بيرفوشين أن العدد أولي و هذا لم يذكره ميرسين ، كذلك أثبت باورس (Powers ) في بداية القرن العشرين أن ميرسين أغفل أيضا العددين الأوليين و و بنهاية عام 1947 كانت سلسلة ميرسين للأعداد (n<258 ) قد اكتملت بشكلها الصحيح و هي :

(n = 2,3,5,7,13,17,19,31,61,89,107,127 ) ، أما بالنسبة لبقية أعداد ميرسين فقد تم اكتشافها مع ظهور الحاسب الحالي.


البحث عن أعداد مرسن الأولية

Graph of number of digits in largest known Mersenne prime by year – electronic era. Note that the vertical scale, the number of digits, is a double logarithmic scale of the value of the prime.


نظريات عن أعداد مرسن

  1. If a and p are natural numbers such that ap − 1 is prime, then a = 2 or p = 1.
    • Proof: a ≡ 1 (mod a − 1). Then ap ≡ 1 (mod a − 1), so ap − 1 ≡ 0 (mod a − 1). Thus a − 1 | ap − 1. However, ap − 1 is prime, so a − 1 = ap − 1 or a − 1 = ±1. In the former case, a = ap, hence a = 0,1 (which is a contradiction, as neither 1 nor 0 is prime) or p = 1. In the latter case, a = 2 or a = 0. If a = 0, however, 0p − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2.
  2. If 2p - 1 is prime, then p is prime.
    • Proof: suppose that p is composite, hence can be written p = a⋅b with a and b > 1. Then (2a)b − 1 is prime, but b > 1 and 2a > 2, contradicting statement 1.
  3. If p is an odd prime, then any prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 1 is prime.
    • Examples: Example I: 25 − 1 = 31 is prime, and 31 = 1 + 3×2×5. Example II: 211 − 1 = 23×89, where 23 = 1 + 2×11, and 89 = 1 + 4×2×11.
    • Proof: If q divides 2p − 1 then 2p ≡ 1 (mod q). By Fermat's Little Theorem, 2(q − 1) ≡ 1 (mod q). Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that (q − 1)(p − 1) ≡ 1 (mod p). Thus there is a number x ≡ (q − 1)(p − 2) for which (q − 1)·x ≡ 1 (mod p), and therefore a number k for which (q − 1)·x − 1 = kp. Since 2(q − 1) ≡ 1 (mod q), raising both sides of the congruence to the power x gives 2(q − 1)x ≡ 1, and since 2p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2kp ≡ 1. Thus 2(q − 1)x/2kp = 2(q − 1)x − kp ≡ 1 (mod q). But by definition, (q − 1)x − kp = 1, implying that 21 ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p and q − 1 are relatively prime is untenable. Since p is prime q − 1 must be a multiple of p. Of course, if the number m = (q − 1)⁄p is odd, then q will be even, since it is equal to mp + 1. But q is prime and cannot be equal to 2; therefore, m is even.
    • Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2p − 1 must be larger than p.
  4. If p is an odd prime, then any prime q that divides must be congruent to ±1 (mod 8).
    • Proof: , so is a square root of 2 modulo . By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to ±1 (mod 8).
  5. A Mersenne prime cannot be a Wieferich prime.
    • Proof: We show if p = 2m - 1 is a Mersenne prime, then the congruence 2p - 1 ≡ 1 does not satisfy. By Fermat's Little theorem, . Now write, . If the given congruence satisfies, then ,therefore 0 ≡ (2mλ - 1)/(2m - 1) = 1 + 2m + 22m + ... + 2λ-1m ≡ -λ mod(2m - 1}. Hence ,and therefore λ ≥ 2m − 1. This leads to p − 1  ≥ m(2m − 1), which is impossible since m ≥ 2.
  6. A prime number divides at most one prime-exponent Mersenne number[2]
  7. If p and 2p+1 are both prime (meaning that p is a Sophie Germain prime), and p is congruent to 3 (mod 4), then 2p+1 divides 2p − 1.
    • Example: 11 and 23 are both prime, and 11 = 2×4+3, so 23 divides 211 − 1.
  8. All composite divisors of prime-exponent Mersenne numbers pass the Fermat primality test for the base 2.

قائمة أعداد مرسن الأولية المعروفة

الجدول التالي لقوائم لأعداد مرسن الأولية المعروفة (المتتالية A000668 في OEIS):

# p Mp Mp digits تاريخ الاكتشاف المكتشف الطريقة المستخدمة
1 2 3 1 ح. 430 ق.م رياضياتيون يونانيون قدماء[3]
2 3 7 1 ح. 430 ق.م Ancient Greek mathematicians[3]
3 5 31 2 c. 300 BCE Ancient Greek mathematicians[4]
4 7 127 3 ح. 300 ق.م Ancient Greek mathematicians[4]
5 13 8191 4 1456 Anonymous[5][6] Trial division
6 17 131071 6 1588[7] Pietro Cataldi Trial division[8]
7 19 524287 6 1588 Pietro Cataldi Trial division[9]
8 31 2147483647 10 1772 Leonhard Euler[10][11] Enhanced trial division[12]
9 61 2305843009213693951 19 1883 November[13] I. M. Pervushin Lucas sequences
10 89 618970019…449562111 27 1911 June[14] R. E. Powers Lucas sequences
11 107 162259276…010288127 33 1914 June 1[15][16] R. E. Powers[17] Lucas sequences
12 127 170141183…884105727 39 1876 January 10[18] Édouard Lucas Lucas sequences
13 521 686479766…115057151 157 1952 January 30[19] Raphael M. Robinson LLT / SWAC
14 607 531137992…031728127 183 1952 January 30[19] Raphael M. Robinson LLT / SWAC
15 1,279 104079321…168729087 386 1952 June 25[20] Raphael M. Robinson LLT / SWAC
16 2,203 147597991…697771007 664 1952 October 7[21] Raphael M. Robinson LLT / SWAC
17 2,281 446087557…132836351 687 1952 October 9[21] Raphael M. Robinson LLT / SWAC
18 3,217 259117086…909315071 969 1957 September 8[22] Hans Riesel LLT / BESK
19 4,253 190797007…350484991 1,281 1961 November 3[23][24] Alexander Hurwitz LLT / IBM 7090
20 4,423 285542542…608580607 1,332 1961 November 3[23][24] Alexander Hurwitz LLT / IBM 7090
21 9,689 478220278…225754111 2,917 1963 May 11[25] Donald B. Gillies LLT / ILLIAC II
22 9,941 346088282…789463551 2,993 1963 May 16[25] Donald B. Gillies LLT / ILLIAC II
23 11,213 281411201…696392191 3,376 1963 June 2[25] Donald B. Gillies LLT / ILLIAC II
24 19,937 431542479…968041471 6,002 1971 March 4[26] Bryant Tuckerman LLT / IBM 360/91
25 21,701 448679166…511882751 6,533 1978 October 30[27] Landon Curt Noll & Laura Nickel LLT / CDC Cyber 174
26 23,209 402874115…779264511 6,987 1979 February 9[28] Landon Curt Noll LLT / CDC Cyber 174
27 44,497 854509824…011228671 13,395 1979 April 8[29][30] Harry Lewis Nelson & David Slowinski LLT / Cray 1
28 86,243 536927995…433438207 25,962 1982 September 25 David Slowinski LLT / Cray 1
29 110,503 521928313…465515007 33,265 1988 January 29[31][32] Walter Colquitt & Luke Welsh LLT / NEC SX-2[33]
30 132,049 512740276…730061311 39,751 1983 September 19[34] David Slowinski LLT / Cray X-MP
31 216,091 746093103…815528447 65,050 1985 September 1[35][36] David Slowinski LLT / Cray X-MP/24
32 756,839 174135906…544677887 227,832 1992 February 17 David Slowinski & Paul Gage LLT / Maple on Harwell Lab Cray-2[37]
33 859,433 129498125…500142591 258,716 1994 January 4[38][39][40] David Slowinski & Paul Gage LLT / Cray C90
34 1,257,787 412245773…089366527 378,632 1996 September 3[41] David Slowinski & Paul Gage[42] LLT / Cray T94
35 1,398,269 814717564…451315711 420,921 1996 November 13 GIMPS / Joel Armengaud[43] LLT / Prime95 on 90 MHz Pentium PC
36 2,976,221 623340076…729201151 895,932 1997 August 24 GIMPS / Gordon Spence[44] LLT / Prime95 on 100 MHz Pentium PC
37 3,021,377 127411683…024694271 909,526 1998 January 27 GIMPS / Roland Clarkson[45] LLT / Prime95 on 200 MHz Pentium PC
38 6,972,593 437075744…924193791 2,098,960 1999 June 1 GIMPS / Nayan Hajratwala[46] LLT / Prime95 on 350 MHz Pentium II IBM Aptiva
39 13,466,917 924947738…256259071 4,053,946 2001 November 14 GIMPS / Michael Cameron[47] LLT / Prime95 on 800 MHz Athlon T-Bird
40 20,996,011 125976895…855682047 6,320,430 2003 November 17 GIMPS / Michael Shafer[48] LLT / Prime95 on 2 GHz Dell Dimension
41 24,036,583 299410429…733969407 7,235,733 2004 May 15 GIMPS / Josh Findley[49] LLT / Prime95 on 2.4 GHz Pentium 4 PC
42 25,964,951 122164630…577077247 7,816,230 2005 February 18 GIMPS / Martin Nowak[50] LLT / Prime95 on 2.4 GHz Pentium 4 PC
43[*] 30,402,457 315416475…652943871 9,152,052 2005 December 15 GIMPS / Curtis Cooper & Steven Boone[51] LLT / Prime95 on 2 GHz Pentium 4 PC
44[*] 32,582,657 124575026…053967871 9,808,358 2006 September 4 GIMPS / Curtis Cooper & Steven Boone[52] LLT / Prime95 on 3 GHz Pentium 4 PC
45[*] 37,156,667 202254406…308220927 11,185,272 2008 September 6 GIMPS / Hans-Michael Elvenich[53] LLT / Prime95 on 2.83 GHz Core 2 Duo PC
46[*] 42,643,801 169873516…562314751 12,837,064 2009 April 12[**] GIMPS / Odd M. Strindmo[54] LLT / Prime95 on 3 GHz Core 2 PC
47[*] 43,112,609 316470269…697152511 12,978,189 2008 August 23 GIMPS / Edson Smith[53] LLT / Prime95 on Dell Optiplex 745
48[*] 57,885,161 581887266…724285951 17,425,170 2013 January 25 GIMPS / Curtis Cooper[55] LLT / Prime95 on 3 GHz Core 2 Duo PC[56]



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تحليل أرقام مرسن المركبة

التعميم

أعداد مرسن في الطبيعة وأماكن أخرى

في معضلة برج هانوا الرياضية: حلحلة المعضلة حيث عدد الأقراص هو p تتطلب على الأقل Mp خطوة.



انظر أيضاً

المصادر

  1. ^ Regius, Hudalricus. Utrisque Arithmetices Epitome.
  2. ^ Will Edgington's Mersenne Page
  3. ^ أ ب There is no mentioning among the ancient egyptians of prime numbers and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BCE) the egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. See Prime Numbers Divide [Retrieved 2012-11-11]. "The egyptians used ($) in the table above for the first primes r=3, 5, 7, or 11 (also for r=23). Here is another intriguing observation: That the egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/n Table [Retrieved 2012-11-11]. In the school of Pythagoras (و. about 570 – d. about 495 BCE) and the pythagoreans we find the first sure observations of prime numbers. Hence the first two Mersenne primes, 3 and 7, were known to and may even be said to have been discovered by them. There is no reference though to their special form 22-1 and 23-1 as such. The sources to the knowledge of prime numbers among the pythagoreans are late. The neoplatonian philosopher Iamblichus, c. 245–c. 325 CE, states that the greek platonian philosopher Speusippus, c. 408 – 339/8 BCE, 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 BCE, who lived a century after Pythagoras, 570 – d. about 495 BCE. In his Theology of Arithmetic in the chapter On the Decad, Iamblichus writes: "Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book On Pythagorean Numbers. In the first half of the book, he elegantly expounds linear numbers [i.e. prime numbers], polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity." Iamblichus The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 112f. [Retrieved 2012-11-11]. In his comments to Nicomachus of Gerasas Introduction to Arithmetic Iamblichus also mentions that Thymaridas, ca. 400 BCE – ca. 350 BCE, uses the term rectilinear for prime numbers and Theon of Smyrna, fl. 100 CE, uses euthymetric and linear as alternative terms. Nicomachus of Gerasa, Introduction to Aritmetic, 1926, p. 127 [Retrieved 2012-11-11] It is unclear though when this said Thymaridas lived. "In a highly suspect passage in Iamblichus, Thymaridas is listed as a pupil of Pythagoras himself." Pythagoreanism [Retrieved 2012-11-11] Before Philolaus, c. 470–c. 385 BCE, we don't have any proof of any knowledge of prime numbers.
  4. ^ أ ب Euclid's Elements, Book IX, Proposition 36
  5. ^ The Prime Pages, Mersenne Primes: History, Theorems and Lists.
  6. ^ We find the oldest (undisputed) note of the result in Codex nr. 14908, which origins from Bibliotheca monasterii ord. S. Benedicti ad S. Emmeramum Ratisbonensis now in the archive of the Bayerische Staatsbibliothek, see "Halm, Karl / Laubmann, Georg von / Meyer, Wilhelm: Catalogus codicum latinorum Bibliothecae Regiae Monacensis, Bd.: 2,2, Monachii, 1876, p. 250". [retrieved on 2012-09-17] The Codex nr. 14908 consists of 10 different medieval works on mathematics and related subjects. The authors of most of these writings are known. Some authors consider the monk Fridericus Gerhart (Amman), c. 1400-d. 1465 (Frater Fridericus Gerhart monachus ordinis sancti Benedicti astrologus professus in monasterio sancti Emmerani diocesis Ratisponensis et in ciuitate eiusdem) to be the author of the part where the prime number 8191 is mentioned. Geschichte Der Mathematik [retrieved on 2012-09-17] The second manuscript of Codex nr. 14908 has the name "Regulae et exempla arithmetica, algebraica, geometrica" and the 5th perfect number and all is factors, including 8191, are mentioned on folio no. 34 a tergo (backside of p. 34). Parts of the manuscript have been published in Archiv der Mathematik und Physik, 13 (1895), pp. 388-406 [retrieved on 2012-09-23]
  7. ^ "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#
  8. ^ p. 13-18 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  9. ^ p. 18-22 in Trattato de' nvumeri perfetti Di Pietro Antonio Cataldo 1603. http://fermi.imss.fi.it/rd/bdv?/bdviewer@selid=1373775#
  10. ^ 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.
  11. ^ http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.
  12. ^ Chris K. Caldwell. "Modular restrictions on Mersenne divisors". Primes.utm.edu. Retrieved 2011-05-21.
  13. ^ “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), p. 553-554. See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48
  14. ^ http://www.jstor.org/stable/2972574 The American Mathematical Monthly, Vol. 18, No. 11 (Nov., 1911), pp. 195-197. The article is signed "DENVER, COLORADO, June, 1911". Retrieved 2011-10-02.
  15. ^ "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, p. 112-119 http://primes.utm.edu/mersenne/LukeMirror/lit/lit_008s.htm [retrieved 2012-10-13]
  16. ^ 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.
  17. ^ The Prime Pages, M107: Fauquembergue or Powers?.
  18. ^ 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.
  19. ^ أ ب "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]
  20. ^ "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]
  21. ^ أ ب "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]
  22. ^ "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]
  23. ^ أ ب 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.
  24. ^ أ ب "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), p. 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]
  25. ^ أ ب ت "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), p. 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]
  26. ^ "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), p. 2319-2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]
  27. ^ "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), p. 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]
  28. ^ "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), p. 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]
  29. ^ David Slowinski, "Searching for the 27th Mersenne Prime", Journal of Recreational Mathematics, v. 11(4), 1978-79, p. 258-261, MR 80g #10013
  30. ^ "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.
  31. ^ "An FFT containing 8192 complex elements, which was the minimum size required to test M110503, ran aproximately 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), p. 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]
  32. ^ "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, p. 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]
  33. ^ "Mersenne Prime Numbers". Omes.uni-bielefeld.de. 2011-01-05. Retrieved 2011-05-21.
  34. ^ "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]
  35. ^ "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. 199-199. 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]
  36. ^ "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]
  37. ^ The Prime Pages, The finding of the 32nd Mersenne.
  38. ^ Chris Caldwell, The Largest Known Primes.
  39. ^ Crays press release
  40. ^ Slowinskis email
  41. ^ Silicon Graphics' press release http://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]
  42. ^ The Prime Pages, A Prime of Record Size! 21257787-1.
  43. ^ GIMPS Discovers 35th Mersenne Prime.
  44. ^ GIMPS Discovers 36th Known Mersenne Prime.
  45. ^ GIMPS Discovers 37th Known Mersenne Prime.
  46. ^ GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award.
  47. ^ GIMPS, Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid.
  48. ^ GIMPS, Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid.
  49. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 224,036,583-1.
  50. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 225,964,951-1.
  51. ^ GIMPS, Mersenne.org Project Discovers New Largest Known Prime Number, 230,402,457-1.
  52. ^ GIMPS, Mersenne.org Project Discovers Largest Known Prime Number, 232,582,657-1.
  53. ^ أ ب Titanic Primes Raced to Win $100,000 Research Award. Retrieved on 2008-09-16.
  54. ^ "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]
  55. ^ خطأ استشهاد: وسم <ref> غير صحيح؛ لا نص تم توفيره للمراجع المسماة m48
  56. ^ Woltman, George. "NEW MERSENNE PRIME! TOTALLY MERSENNE THIS TIME! thread". mersenneforum. Retrieved 5 February 2013. {{cite web}}: Italic or bold markup not allowed in: |publisher= (help)

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