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| | == Business intelligence Oakley Prescription Sunglasses == |
| | named_after = [[Marin Mersenne]]
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| | publication_year = 1536<ref>{{Cite book | last = Regius | first = Hudalricus | title = Utrisque Arithmetices Epitome | url = http://books.google.de/books?id=hs85AAAAcAAJ&printsec=frontcover&dq=Utriusque+Arithmetices+epitome&hl=de&ei=o4cDTb10y_WyBur_8PkJ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCoQ6AEwAA#v=onepage&q=2047&f=false }}</ref>
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| | author = Regius, H.
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| | terms_number = 48
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| | con_number = Infinite
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| | parentsequence = Mersenne numbers
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| | first_terms = [[3 (number)|3]], [[7 (number)|7]], [[31 (number)|31]], [[127 (number)|127]]
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| | largest_known_term = 2<sup>57885161</sup> − 1 (January 2013)
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| | OEIS = A000668
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| }}
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| In [[mathematics]], a '''Mersenne prime''' is a [[prime number]] of the form <math>M_n=2^n-1</math>. They are named after the French monk [[Marin Mersenne]] who studied them in the early 17th century. The first four Mersenne primes are 3, 7, 31 and 127.
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| If ''n'' is a [[composite number]] then so is 2<sup>''n''</sup> − 1. The definition is therefore unchanged when written <math>M_p=2^p-1</math> where ''p'' is assumed prime.
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| More generally, numbers of the form <math>M_n=2^n-1\,</math> without the primality requirement are called '''Mersenne numbers'''. Mersenne numbers are sometimes defined to have the additional requirement that ''n'' be prime, equivalently that they be [[pernicious number|pernicious]] Mersenne numbers, namely those pernicious numbers whose binary representation contains no zeros. The smallest composite pernicious Mersenne number is 2<sup>11</sup> - 1.
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| {{As of|2014|January|url=http://primes.utm.edu/top20/page.php?id=4}}, 48 Mersenne primes are known. The [[largest known prime number]] <span class=nowrap>(2<sup>57,885,161</sup> − 1)</span> is a Mersenne prime.<ref name="m48">{{cite web|url=http://mersenne.org/various/57885161.htm|title=GIMPS Project Discovers Largest Known Prime Number, 2<sup>57,885,161</sup>-1|publisher=''[[Great Internet Mersenne Prime Search]]''|accessdate=2013-02-05}}</ref><ref>{{cite news|url=http://www.newscientist.com/article/dn23138-new-17milliondigit-monster-is-largest-known-prime.html|title=New 17-million-digit monster is largest known prime|last=Aron|first=Jacob|date=February 5, 2013|work=[[New Scientist]]|accessdate=5 February 2013}}</ref> Since 1997, all newly found Mersenne primes have been discovered by the “[[Great Internet Mersenne Prime Search]]” (GIMPS), a [[distributed computing]] project on the Internet.
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| ==About Mersenne primes==
| | <li>[http://www.cctv0534.com/forum.php?mod=viewthread&tid=12223 http://www.cctv0534.com/forum.php?mod=viewthread&tid=12223]</li> |
| {{unsolved|mathematics|Are there infinitely many Mersenne primes?}}
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| | | <li>[http://www.sebalo.info/spip/spip.php?article13 http://www.sebalo.info/spip/spip.php?article13]</li> |
| Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The [[Lenstra–Pomerance–Wagstaff conjecture]] asserts that there are infinitely many Mersenne primes and predicts their [[asymptotic analysis|order of growth]]. It is also not known whether infinitely many Mersenne numbers with prime exponents are [[composite number|composite]], although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of [[Sophie Germain prime]]s [[Congruence relation|congruent]] to 3 ([[Modular arithmetic|mod 4]]).
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| | | <li>[http://dag.he4136.vps.webenabled.net/content/parish-registration-0#comment-156604 http://dag.he4136.vps.webenabled.net/content/parish-registration-0#comment-156604]</li> |
| The first four Mersenne primes are
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| | | <li>[http://124.128.87.5:8888/discuz/forum.php?mod=viewthread&tid=437446 http://124.128.87.5:8888/discuz/forum.php?mod=viewthread&tid=437446]</li> |
| : ''M''<sub>2</sub> = 3, ''M''<sub>3</sub> = 7, ''M''<sub>5</sub> = 31 and ''M''<sub>7</sub> = 127.
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| | | </ul> |
| A basic [[theorem]] about Mersenne numbers states that if ''M''<sub>''p''</sub> is prime, then the exponent ''p'' must also be prime. This follows from the identity
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| :<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>
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| This rules out primality for Mersenne numbers with composite exponent, such as ''M''<sub>4</sub> = 2<sup>4</sup> − 1 = 15 = 3×5 = (2<sup>2</sup> − 1)×(1 + 2<sup>2</sup>).
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| Though it was believed by early mathematicians that ''M''<sub>''p''</sub> is prime for all primes ''p'', ''M''<sub>''p''</sub> is very rarely prime. In fact, of the 1,622,441 prime numbers ''p'' up to 25,964,951,<ref>{{cite web |url=http://www.wolframalpha.com/input/?i=Number+of+primes+%3C%3D+25964951 |title=Number of primes <= 25964951 |publisher=[[Wolfram Alpha]] |accessdate=2013-03-27}}</ref> ''M''<sub>''p''</sub> is prime for only 42 of them. The smallest counterexample is the Mersenne number
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| : ''M''<sub>11</sub> = 2<sup>11</sup> − 1 = 2047 = 23 × 89.
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| The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The [[Lucas–Lehmer primality test]] (LLT) is an efficient [[primality test]] that greatly aids this task. The search for the largest known prime has somewhat of a [[cult following]]. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using [[distributed computing]].
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| Mersenne primes are used in [[pseudorandom number generator]]s such as the [[Mersenne twister]], [[Park–Miller random number generator]], Generalized Shift Register and Fibonacci RNG.
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| ==Perfect numbers==
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| Mersenne primes ''M''<sub>''p''</sub> are also noteworthy due to their connection to [[perfect number]]s. In the 4th century BC, [[Euclid]] proved that if 2<sup>''p''</sup>−1 is prime, then 2<sup>''p''−1</sup>(2<sup>''p''</sup>−1) is a perfect number. This number, also expressible as ''M''<sub>''p''</sub>(''M''<sub>''p''</sub>+1)/2, is the ''M''<sub>''p''</sub>th [[triangular number]] and the 2<sup>''p''−1</sup>th [[hexagonal number]]. In the 18th century, [[Leonhard Euler]] proved that, conversely, all even perfect numbers have this form.<ref>Chris K. Caldwell, [http://primes.utm.edu/mersenne/index.html Mersenne Primes: History, Theorems and Lists]</ref> It is unknown whether there are any [[odd perfect number]]s.
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| ==History==
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| Mersenne primes take their name from the 17th-century [[France|French]] scholar [[Marin Mersenne]], who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. His list was largely incorrect, as Mersenne mistakenly included ''M''<sub>67</sub> and ''M''<sub>257</sub> (which are composite), and omitted ''M''<sub>61</sub>, ''M''<sub>89</sub>, and ''M''<sub>107</sub> (which are prime). Mersenne gave little indication how he came up with his list.<ref>The Prime Pages, [http://primes.utm.edu/glossary/page.php?sort=MersennesConjecture Mersenne's conjecture].</ref>
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| [[Édouard Lucas]] proved in 1876 that ''M''<sub>127</sub> is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever calculated by hand. ''M''<sub>61</sub> was determined to be prime in 1883 by [[Ivan Mikheevich Pervushin]], though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M<sub>67</sub> is actually composite. No factor was found until a famous talk by [[Frank Nelson Cole|Cole]] in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking.<ref>{{cite book
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| |title=Mathematics, queen and servant of science
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| |author=Bell, E.T. and Mathematical Association of America
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| |year=1951
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| |publisher=McGraw-Hill New York}} p. 228.</ref> He later said that the result had taken him "three years of Sundays" to find.<ref>http://news.bbc.co.uk/dna/place-lancashire/plain/A670051</ref> A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.
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| ==Searching for Mersenne primes==
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| Fast algorithms for finding Mersenne primes are available, and as of 2014 the ten [[largest known prime number]]s are Mersenne primes.
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| The first four Mersenne primes ''M''<sub>2</sub> = 3, ''M''<sub>3</sub> = 7, ''M''<sub>5</sub> = 31 and M<sub>7</sub> = 127 were known in antiquity. The fifth, ''M''<sub>13</sub> = 8191, was discovered anonymously before 1461; the next two (''M''<sub>17</sub> and ''M''<sub>19</sub>) were found by [[Pietro Cataldi|Cataldi]] in 1588. After nearly two centuries, ''M''<sub>31</sub> was verified to be prime by [[Leonhard Euler|Euler]] in 1772. The next (in historical, not numerical order) was ''M''<sub>127</sub>, found by [[Édouard Lucas|Lucas]] in 1876, then ''M''<sub>61</sub> by [[Ivan Mikheevich Pervushin|Pervushin]] in 1883. Two more (''M''<sub>89</sub> and ''M''<sub>107</sub>) were found early in the 20th century, by [[R. E. Powers|Powers]] in 1911 and 1914, respectively.
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| The best method presently known for testing the primality of Mersenne numbers is the [[Lucas–Lehmer primality test]]. Specifically, it can be shown that for prime ''p'' > 2, ''M<sub>p''</sub> = 2<sup>''p''</sup> − 1 is prime if and only if ''M<sub>p''</sub> divides ''S<sub>p''−2</sub>, where ''S''<sub>0</sub> = 4 and, for ''k'' > 0,
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| :<math>S_k = S_{k-1}^2-2.\ </math>
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| [[Image:Digits in largest Mersenne prime by year.svg|thumb|right|400px|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.]]
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| The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. [[Alan Turing]] searched for them on the [[Manchester Mark 1]] in 1949,<ref>Brian Napper, [http://www.computer50.org/mark1/maths.html The Mathematics Department and the Mark 1].</ref> but the first successful identification of a Mersenne prime, ''M''<sub>521</sub>, by this means was achieved at 10:00 pm on January 30, 1952 using the U.S. [[National Bureau of Standards]] [[SWAC (computer)|Western Automatic Computer (SWAC)]] at the Institute for Numerical Analysis at the [[University of California, Los Angeles]], under the direction of [[Derrick Henry Lehmer|Lehmer]], with a computer search program written and run by Prof. [[Raphael M. Robinson|R. M. Robinson]]. It was the first Mersenne prime to be identified in thirty-eight years; the next one, ''M''<sub>607</sub>, was found by the computer a little less than two hours later. Three more — ''M''<sub>1279</sub>, ''M''<sub>2203</sub>, ''M''<sub>2281</sub> — were found by the same program in the next several months. ''M''<sub>4253</sub> is the first Mersenne prime that is [[titanic prime|titanic]], ''M''<sub>44497</sub> is the first [[gigantic prime|gigantic]], and ''M''<sub>6,972,593</sub> was the first [[megaprime]] to be discovered, being a prime with at least 1,000,000 digits.<ref>The Prime Pages, [http://primes.utm.edu/glossary/page.php?sort=Megaprime The Prime Glossary: megaprime].</ref> All three were the first known prime of any kind of that size.
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| In September 2008, mathematicians at [[University of California, Los Angeles|UCLA]] participating in GIMPS won part of a $100,000 prize from the [[Electronic Frontier Foundation]] for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a [[Dell OptiPlex]] 745 on August 23, 2008. This is the eighth Mersenne prime discovered at UCLA.<ref>{{cite web|url=http://www.latimes.com/news/science/la-sci-prime27-2008sep27,0,2746766.story |title=UCLA mathematicians discover a 13-million-digit prime number|publisher=Los Angeles Times|date=2008-09-27 |accessdate=2011-05-21}}</ref>
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| On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. This report was apparently overlooked until June 4, 2009. The find was verified on June 12, 2009. The prime is 2{{sup|42,643,801}} − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered.
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| On January 25, 2013, [[Curtis Cooper (mathematician)|Curtis Cooper]], a mathematician at the [[University of Central Missouri]], discovered a 48th Mersenne prime, 2{{sup|57,885,161}} − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.<ref>{{cite web| url=http://www.scientificamerican.com/article.cfm?id=largest-prime-number-disc |title=Largest Prime Number Discovered |author=Tia Ghose |publisher=[[Scientific American]] |accessdate=2013-02-07}}</ref> This was the third Mersenne prime discovered by Dr. Cooper and his team in the past seven years.
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| The Electronic Frontier Foundation (EFF) offers a prize of $150,000 to the first individual or group who discovers a prime number with at least 100,000,000 decimal digits<ref>https://www.eff.org/awards/coop/</ref> (the smallest Mersenne number with said amount of digits is 2<sup>''332192807''</sup> − 1).
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| ==Theorems about Mersenne numbers==
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| <!-- WARNING: you're NOT going to get a nice display by intermixing HTML-style lists with WIKIMEDIA-style markup within list items ! this has been tried !-->
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| # If ''a'' and ''p'' are natural numbers such that ''a''<sup>''p''</sup> − 1 is prime, then ''a'' = 2 or ''p'' = 1.
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| #* '''Proof''': {{nowrap|''a'' ≡ 1 ([[Modular arithmetic|mod]] ''a'' − 1).}} Then {{nowrap|''a''<sup>''p''</sup> ≡ 1 (mod ''a'' − 1),}} so {{nowrap|''a''<sup>''p''</sup> − 1 ≡ 0 (mod ''a'' − 1).}} Thus {{nowrap|''a'' − 1 {{!}} ''a''<sup>''p''</sup> − 1.}} However, {{nowrap|''a''<sup>''p''</sup> − 1}} is prime, so {{nowrap|''a'' − 1 {{=}} ''a''<sup>''p''</sup> − 1}} or {{nowrap|''a'' − 1 {{=}} ±1.}} In the former case, {{nowrap|''a'' {{=}} ''a<sup>p</sup>'',}} hence {{nowrap|''a'' {{=}} 0,1}} (which is a contradiction, as neither 1 nor 0 is prime) or {{nowrap|''p'' {{=}} 1.}} In the latter case, {{nowrap|''a'' {{=}} 2}} or {{nowrap|''a'' {{=}} 0.}} If {{nowrap|''a'' {{=}} 0,}} however, {{nowrap|0<sup>''p''</sup> − 1 {{=}} 0 − 1 {{=}} −1}} which is not prime. Therefore, {{nowrap|''a'' {{=}} 2.}}
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| # If 2<sup>''p''</sup> − 1 is prime, then ''p'' is prime.
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| #* '''Proof''': suppose that ''p'' is composite, hence can be written {{nowrap|''p'' {{=}} ''a''⋅''b''}} with ''a'' and {{nowrap|''b'' > 1.}} Then 2<sup>''p''</sup> - 1 = 2<sup>''ab''</sup> - 1 = (2<sup>''a''</sup>)<sup>''b''</sup> - 1 = (2<sup>''a''</sup> - 1)[(2<sup>''a''</sup>)<sup>b - 1</sup> + (2<sup>''a''</sup>)<sup>b - 2</sup> + … + 2<sup>''a''</sup> + 1] so 2<sup>''p''</sup> - 1 is composite contradicting our assumption that 2<sup>''p''</sup> − 1 is prime.
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| # If ''p'' is an odd prime, then any prime ''q'' that divides 2<sup>''p''</sup> − 1 must be 1 plus a multiple of 2''p''. This holds even when 2<sup>''p''</sup> − 1 is prime.
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| #* '''Examples''': Example I: 2<sup>5</sup> − 1 = 31 is prime, and 31 = 1 + 3×(2×5). Example II: 2<sup>11</sup> − 1 = 23×89, where 23 = 1 + (2×11), and 89 = 1 + 4×(2×11).
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| #* '''Proof''': By [[Fermat's little theorem]], if ''q'' is a prime factor of 2<sup>''p''</sup> − 1, ''q'' is a factor of 2<sup>''q'' − 1</sup> − 1. Since ''q'' ia a factor of 2<sup>''p''</sup> − 1, for all positive integers ''c'', ''q'' is also a factor of 2<sup>''pc''</sup> − 1. Since ''p'' is prime and ''q'' is not a factor of 2<sup>1</sup> − 1, ''p'' is also the smallest positive integer ''x'' such that ''q'' is a factor of 2<sup>''x''</sup> − 1. As a result, for all positive integers ''x'', ''q'' is a factor of 2<sup>''x''</sup> − 1 if and only if ''p'' is a factor of ''x''. Therefore, since ''q'' is a factor of 2<sup>''q'' − 1</sup> − 1, ''p'' is a factor of ''q'' − 1 so ''q'' ≡ 1 mod ''p''. Furthermore, since ''q'' is prime and greater than 2, ''q'' is odd. Therefore ''q'' ≡ 1 mod 2''p''.
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| #* '''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 2<sup>''p''</sup> − 1 must be larger than ''p''.</li>
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| # If ''p'' is an odd prime, then any prime ''q'' that divides <math>2^p-1</math> must be congruent to ±1 (mod 8).
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| #* '''Proof''': <math>2^{p+1} = 2 \pmod q</math>, so <math>2^{(p+1)/2}</math> is a square root of 2 modulo <math>q</math>. By [[quadratic reciprocity]], any prime modulo which 2 has a square root is congruent to ±1 (mod 8).
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| # A Mersenne prime cannot be a [[Wieferich prime]].
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| #* '''Proof''': We show if ''p'' = 2<sup>''m''</sup> − 1 is a Mersenne prime, then the congruence 2<sup>''p''</sup> − 1 ≡ 1 does not satisfy. By Fermat's Little theorem, <math>m \mid p-1</math>. Now write, <math>p-1=m\lambda</math>. If the given congruence satisfies, then <math>p^2\mid2^{m\lambda}-1</math>,therefore 0 ≡ (2<sup>''m''λ</sup> − 1)/(2<sup>''m''</sup> − 1) = 1 + 2<sup>''m''</sup> + 2<sup>2''m''</sup> + ... + 2<sup>λ−1''m''</sup> ≡ −λ mod(2<sup>''m''</sup> − 1}. Hence <math>2^m-1\mid\lambda</math>,and therefore λ ≥ 2<sup>''m''</sup> − 1. This leads to ''p'' − 1 ≥ ''m''(2<sup>''m''</sup> − 1), which is impossible since ''m'' ≥ 2.
| |
| # A prime number divides at most one prime-exponent Mersenne number<ref>[http://www.garlic.com/~wedgingt/mersenne.html Will Edgington's Mersenne Page]</ref>
| |
| # If ''p'' and 2''p'' + 1 are both prime (meaning that ''p'' is a [[Sophie Germain prime]]), and ''p'' is [[Congruence relation|congruent]] to 3 (mod 4), then 2''p'' + 1 divides 2<sup>''p''</sup> − 1.<ref>[http://primes.utm.edu/notes/proofs/MerDiv2.html Proof of a result of Euler and Lagrange on Mersenne Divisors]</ref>
| |
| #* '''Example''': 11 and 23 are both prime, and 11 = 2×4 + 3, so 23 divides 2<sup>11</sup> − 1.
| |
| # All composite divisors of prime-exponent Mersenne numbers pass the [[Fermat primality test]] for the base 2.
| |
| # The number of digits in the decimal representation of <math>M_n</math> equals <math>\lfloor n\cdot \log_{10}2\rfloor+1</math>, where <math>\lfloor x\rfloor</math> denotes the [[floor function]].
| |
| | |
| ==List of known Mersenne primes==
| |
| The table below lists all known Mersenne primes {{OEIS|id=A000668}}:
| |
| {| class="wikitable"
| |
| |-
| |
| ! #
| |
| ! ''p''
| |
| ! ''M''<sub>''p''</sub>
| |
| ! ''M''<sub>''p''</sub> digits
| |
| ! Discovered
| |
| ! Discoverer
| |
| ! Method used
| |
| |-
| |
| | style="text-align:right;"| 1
| |
| | style="text-align:right;"| 2
| |
| | style="text-align:right;"| [[3 (number)|3]]
| |
| | style="text-align:right;"| 1
| |
| | c. 430 BC
| |
| | [[Greek mathematics|Ancient Greek mathematicians]]<ref name="Philolaus">There is no mentioning among the [[Ancient Egypt|ancient Egyptians]] of prime numbers, and they did not have any concept for prime numbers known today. In the [[Rhind papyrus]] (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. See [http://www.ukessays.com/essays/general-studies/prime-numbers-divide.php 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's Sieve 2000 years before Eratosthenes 'discovered' it." [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptroll2-n.html The Rhind 2/n Table] [Retrieved 2012-11-11].
| |
| | |
| In the school of [[Pythagoras]] (b. about 570 – d. about 495 BC) and the [[Pythagoreans]], we find the first sure observations of prime numbers. Hence the first two Mersenne primes, 3 and 7, were known to and may even be said to have been discovered by them. There is no reference, though, to their special form 2<sup>2</sup>-1 and 2<sup>3</sup>-1 as such.
| |
| | |
| The sources to the knowledge of prime numbers among the Pythagoreans are late. The Neoplatonic philosopher [[Iamblichus]], AD c. 245–c. 325, states that the Greek Platonic philosopher [[Speusippus]], c. 408 – 339/8 BC, wrote a book named ''On Pythagorean Numbers''. According to Iamblichus this book was based on the works of the Pythagorean [[Philolaus]], c. 470–c. 385 BC, who lived a century after [[Pythagoras]], 570 – c. 495 BC. In his ''Theology of Arithmetic'' in the chapter ''On the Decad'', Iamblichus writes: "Speusippus, the son of Plato's sister Potone, and head of the Academy before Xenocrates, compiled a polished little book from the Pythagorean writings which were particularly valued at any time, and especially from the writings of Philolaus; he entitled the book ''On Pythagorean Numbers''. In the first half of the book, he elegantly expounds linear numbers [i.e. prime numbers], polygonal numbers and all sorts of plane numbers, solid numbers and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their proportionality and reciprocity." [http://www.scribd.com/doc/38568451/Theologoumena-Arithmeticae#page=112 ''Iamblichus'' The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 112f.] [Retrieved 2012-11-11].
| |
| | |
| [[Iamblichus]] also gives us a direct quote from [[Speusippus]]' book where [[Speusippus]] among other things writes: "Secondly, it is necessary for a perfect number [the concept "perfect number" is not used here in a modern sence] to contain an equal amount of prime and incomposite numbers, and secondary and composite numbers." [http://www.scribd.com/doc/38568451/Theologoumena-Arithmeticae#page=113 ''Iamblichus'' The Theology of Arithmetic translated by Robin Waterfiled, 1988, p. 113.] [Retrieved 2012-11-11]. For the Greek original text, see [http://books.google.se/books?id=cUPXqSb7V1wC&lpg=PA276&ots=Q3QhUOGtvH&dq=Speusippus%20prime&hl=sv&pg=PA140#v=onepage&q&f=false Speusippus of Athens: A Critical Study with a Collection of the Related Texts and Commentary by Leonardo Tarán, 1981, p. 140 line 21–22] [Retrieved 2012-11-11]
| |
| | |
| In his comments to [[Nicomachus|Nicomachus of Gerasas]]'s [[Introduction to Arithmetic]], [[Iamblichus]] also mentions that [[Thymaridas]], ca. 400 BC – ca. 350 BC, uses the term ''rectilinear'' for prime numbers, and that [[Theon of Smyrna]], fl. AD 100, uses ''euthymetric'' and ''linear'' as alternative terms. [http://ia700709.us.archive.org//load_djvu_applet.php?file=27/items/NicomachusIntroToArithmetic/nicomachus_introduction_arithmetic.djvu Nicomachus of Gerasa, Introduction to 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." [http://plato.stanford.edu/entries/pythagoreanism/#hippasus Pythagoreanism] [Retrieved 2012-11-11]
| |
| | |
| Before [[Philolaus]], c. 470–c. 385 BC, we don't have any proof of any knowledge of prime numbers.</ref>
| |
| |
| |
| |-
| |
| | style="text-align:right;"| 2
| |
| | style="text-align:right;"| 3
| |
| | style="text-align:right;"| [[7 (number)|7]]
| |
| | style="text-align:right;"| 1
| |
| | c. 430 BC
| |
| | Ancient Greek mathematicians<ref name="Philolaus"/>
| |
| |
| |
| |-
| |
| | style="text-align:right;"| 3
| |
| | style="text-align:right;"| 5
| |
| | style="text-align:right;"| [[31 (number)|31]]
| |
| | style="text-align:right;"| 2
| |
| | c. 300 BC
| |
| | Ancient Greek mathematicians<ref name="Euclid">[http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html Euclid's Elements, Book IX, Proposition 36]</ref>
| |
| |
| |
| |-
| |
| | style="text-align:right;"| 4
| |
| | style="text-align:right;"| 7
| |
| | style="text-align:right;"| [[127 (number)|127]]
| |
| | style="text-align:right;"| 3
| |
| | c. 300 BC
| |
| | Ancient Greek mathematicians<ref name="Euclid"/>
| |
| |
| |
| |-
| |
| | style="text-align:right;"| 5
| |
| | style="text-align:right;"| 13
| |
| | style="text-align:right;"| 8191
| |
| | style="text-align:right;"| 4
| |
| | 1456
| |
| | Anonymous<ref name="primepages">The Prime Pages, [http://primes.utm.edu/mersenne/ Mersenne Primes: History, Theorems and Lists].</ref><ref>We find the oldest (undisputed) note of the result in Codex nr. 14908, which origins from Bibliotheca monasterii ord. S. Benedicti ad S. Emmeramum Ratisbonensis now in the archive of the Bayerische Staatsbibliothek, see "Halm, Karl / Laubmann, Georg von / [http://daten.digitale-sammlungen.de/~db/bsb00008253/images/index.html?fip=193.174.98.30&seite=254 Meyer, Wilhelm: Catalogus codicum latinorum Bibliothecae Regiae Monacensis, Bd.: 2,2, Monachii, 1876, p. 250".] [retrieved on 2012-09-17] The Codex nr. 14908 consists of 10 different medieval works on mathematics and related subjects. The authors of most of these writings are known. Some authors consider the monk Fridericus Gerhart (Amman), 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. [http://audio02.archive.org/stream/vorlesungenber02cantuoft#page/238/mode/2up Geschichte Der Mathematik] [retrieved on 2012-09-17] The second manuscript of Codex nr. 14908 has the name "Regulae et exempla arithmetica, algebraica, geometrica" and the 5th perfect number and all is factors, including 8191, are mentioned on folio no. 34 a tergo (backside of p. 34). Parts of the manuscript have been published in [http://archive.org/stream/archivdermathem91unkngoog#page/n389/mode/2up Archiv der Mathematik und Physik, 13 (1895), pp. 388–406] [retrieved on 2012-09-23]</ref>
| |
| | [[Trial division]]
| |
| |-
| |
| | style="text-align:right;"| 6
| |
| | style="text-align:right;"| 17
| |
| | style="text-align:right;"| 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#</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#</ref>
| |
| | |
| |-
| |
| | style="text-align:right;"| 7
| |
| | style="text-align:right;"| 19
| |
| | style="text-align:right;"| 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#</ref>
| |
| |-
| |
| | style="text-align:right;"| 8
| |
| | style="text-align:right;"| 31
| |
| | style="text-align:right;"| [[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 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.</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
| |
| | |
| 2<sup>61</sup> – 1 = 2305843009213693951
| |
| | |
| est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. http://www.biodiversitylibrary.org/item/107789#page/277/mode/1up [retrieved 2012-09-17]
| |
| | |
| See also Mélanges mathématiques et astronomiques tirés du Bulletin de l’Académie impériale des sciences de St.-Pétersbourg v. 6 (1881–1888), pp. 553–554.
| |
| | |
| See also Mémoires de l'Académie impériale des sciences de St.-Pétersbourg: Sciences mathématiques, physiques et naturelles, vol. 48
| |
| </ref>
| |
| | [[Ivan Mikheevich Pervushin|I. M. Pervushin]]
| |
| | [[Lucas sequence]]s
| |
| |-
| |
| | style="text-align:right;"| 10
| |
| | style="text-align:right;"| 89
| |
| | style="text-align:right;"| 618970019…449562111
| |
| | style="text-align:right;"| 27
| |
| | 1911 June<ref>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.</ref>
| |
| | [[R. E. Powers]]
| |
| | Lucas sequences
| |
| |-
| |
| | style="text-align:right;"| 11
| |
| | style="text-align:right;"| 107
| |
| | style="text-align:right;"| 162259276…010288127
| |
| | style="text-align:right;"| 33
| |
| | 1914 June 1<ref>"M. E. Fauquenbergue a trouvé ses résultats depuis Février, et j’en ai reçu communication le 7 Juin; M. Powers a envoyé le 1<sup>er</sup> Juin un cablógramme à [[Thomas John I'Anson Bromwich|M. Bromwich]] [secretary of London Mathematical Society] pour 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>
| |
| | [[R. E. Powers]]<ref>The Prime Pages, [http://primes.utm.edu/notes/fauquem.html M<sub>107</sub>: Fauquembergue or Powers?].</ref>
| |
| | Lucas sequences
| |
| |-
| |
| | style="text-align:right;"| 12
| |
| | style="text-align:right;"| 127
| |
| | style="text-align:right;"| 170141183…884105727
| |
| | style="text-align:right;"| 39
| |
| | 1876 January 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;"| 686479766…115057151
| |
| | style="text-align:right;"| 157
| |
| | 1952 January 30<ref name=autogenerated4>"Using the standard Lucas test for Mersenne primes as programmed by R. M. Robinson, the SWAC has discovered the primes 2<sup>521</sup> − 1 and 2<sup>607</sup> − 1 on January 30, 1952." D. H. Lehmer, ''Recent Discoveries of Large Primes'', Mathematics of Computation, vol. 6, No. 37 (1952), p. 61, http://www.ams.org/journals/mcom/1952-06-037/S0025-5718-52-99404-0/S0025-5718-52-99404-0.pdf [Retrieved 2012-09-18]</ref>
| |
| | [[Raphael M. Robinson]]
| |
| | [[Lucas–Lehmer primality test|LLT]] / [[SWAC (computer)|SWAC]]
| |
| |-
| |
| | style="text-align:right;"| 14
| |
| | style="text-align:right;"| 607
| |
| | style="text-align:right;"| 531137992…031728127
| |
| | style="text-align:right;"| 183
| |
| | 1952 January 30<ref name=autogenerated4 />
| |
| | Raphael M. Robinson
| |
| | LLT / SWAC
| |
| |-
| |
| | style="text-align:right;"| 15
| |
| | style="text-align:right;"| 1,279
| |
| | style="text-align:right;"| 104079321…168729087
| |
| | style="text-align:right;"| 386
| |
| | 1952 June 25<ref>"The program described in Note 131 (c) has produced the 15th Mersenne prime 2<sup>1279</sup> − 1 on June 25. The SWAC tests this number in 13 minutes and 25 seconds." D. H. Lehmer, ''A New Mersenne Prime'', Mathematics of Computation, vol. 6, No. 39 (1952), p. 205, http://www.ams.org/journals/mcom/1952-06-039/S0025-5718-52-99387-3/S0025-5718-52-99387-3.pdf [Retrieved 2012-09-18]</ref>
| |
| | Raphael M. Robinson
| |
| | LLT / SWAC
| |
| |-
| |
| | style="text-align:right;"| 16
| |
| | style="text-align:right;"| 2,203
| |
| | style="text-align:right;"| 147597991…697771007
| |
| | style="text-align:right;"| 664
| |
| | 1952 October 7<ref name=autogenerated2>"Two more Mersenne primes, 2<sup>2203</sup> − 1 and 2<sup>2281</sup> − 1, were discovered by the SWAC on October 7 and 9, 1952." D. H. Lehmer, ''Two New Mersenne Primes'', Mathematics of Computation, vol. 7, No. 41 (1952), p. 72, http://www.ams.org/journals/mcom/1953-07-041/S0025-5718-53-99371-5/S0025-5718-53-99371-5.pdf [Retrieved 2012-09-18]</ref>
| |
| | Raphael M. Robinson
| |
| | LLT / SWAC
| |
| |-
| |
| | style="text-align:right;"| 17
| |
| | style="text-align:right;"| 2,281
| |
| | style="text-align:right;"| 446087557…132836351
| |
| | style="text-align:right;"| 687
| |
| | 1952 October 9<ref name=autogenerated2 />
| |
| | Raphael M. Robinson
| |
| | LLT / SWAC
| |
| |-
| |
| | style="text-align:right;"| 18
| |
| | style="text-align:right;"| 3,217
| |
| | style="text-align:right;"| 259117086…909315071
| |
| | style="text-align:right;"| 969
| |
| | 1957 September 8<ref>"On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number M<sub>3217</sub> = 2<sup>3217</sup> − 1 is a prime." Hans Riesel, ''A New Mersenne Prime'', Mathematics of Computation, vol. 12 (1958), p. 60, http://www.ams.org/journals/mcom/1958-12-061/S0025-5718-1958-0099752-6/S0025-5718-1958-0099752-6.pdf [Retrieved 2012-09-18]</ref>
| |
| | [[Hans Riesel]]
| |
| | LLT / [[BESK]]
| |
| |-
| |
| | style="text-align:right;"| 19
| |
| | style="text-align:right;"| 4,253
| |
| | style="text-align:right;"| 190797007…350484991
| |
| | style="text-align:right;"| 1,281
| |
| | 1961 November 3<ref name=autogenerated1>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 ''p'' is prime, ''M''<sub>''p</sub> = 2<sup>''p''</sup> − 1 is called a Mersenne number. The primes ''M''<sub>4253</sub> and ''M''<sub>4423</sub> were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, ''New Mersenne Primes'', Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18]</ref>
| |
| | Alexander Hurwitz
| |
| | LLT / [[IBM 7090]]
| |
| |-
| |
| | style="text-align:right;"| 20
| |
| | style="text-align:right;"| 4,423
| |
| | style="text-align:right;"| 285542542…608580607
| |
| | style="text-align:right;"| 1,332
| |
| | 1961 November 3<ref name=autogenerated1 /><ref name=autogenerated5 />
| |
| | Alexander Hurwitz
| |
| | LLT / IBM 7090
| |
| |-
| |
| | style="text-align:right;"| 21
| |
| | style="text-align:right;"| 9,689
| |
| | style="text-align:right;"| 478220278…225754111
| |
| | style="text-align:right;"| 2,917
| |
| | 1963 May 11<ref name=autogenerated3>"The primes M<sub>9689</sub>, M<sub>9941</sub>, and M<sub>11213</sub> which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois." Donald B. Gillies, ''Three New Mersenne Primes and a Statistical Theory'', Mathematics of Computation, vol. 18, No. 85 (1964), pp. 93–97, http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0159774-6/S0025-5718-1964-0159774-6.pdf [Retrieved 2012-09-18]</ref>
| |
| | [[Donald B. Gillies]]
| |
| | LLT / [[ILLIAC II]]
| |
| |-
| |
| | style="text-align:right;"| 22
| |
| | style="text-align:right;"| 9,941
| |
| | style="text-align:right;"| 346088282…789463551
| |
| | style="text-align:right;"| 2,993
| |
| | 1963 May 16<ref name=autogenerated3 />
| |
| | Donald B. Gillies
| |
| | LLT / ILLIAC II
| |
| |-
| |
| | style="text-align:right;"| 23
| |
| | style="text-align:right;"| 11,213
| |
| | style="text-align:right;"| 281411201…696392191
| |
| | style="text-align:right;"| 3,376
| |
| | 1963 June 2<ref name=autogenerated3 />
| |
| | Donald B. Gillies
| |
| | LLT / ILLIAC II
| |
| |-
| |
| | style="text-align:right;"| 24
| |
| | style="text-align:right;"| 19,937
| |
| | style="text-align:right;"| 431542479…968041471
| |
| | style="text-align:right;"| 6,002
| |
| | 1971 March 4<ref>"On the evening of March 4, 1971, a zero Lucas-Lehmer residue for p = p<sub>24</sub> = 19937 was found. Hence, M<sub>19937</sub> is the 24th Mersenne prime." Bryant Tuckerman, ''The 24th Mersenne Prime'', Proceedings of the National Academy of Sciences of the United States of America, vol. 68:10 (1971), pp. 2319–2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]</ref>
| |
| | [[Bryant Tuckerman]]
| |
| | LLT / [[IBM 360]]/91
| |
| |-
| |
| | style="text-align:right;"| 25
| |
| | style="text-align:right;"| 21,701
| |
| | style="text-align:right;"| 448679166…511882751
| |
| | style="text-align:right;"| 6,533
| |
| | 1978 October 30<ref>"On October 30, 1978 at 9:40 pm, we found M<sub>21701</sub> to be prime. The CPU time required for this test was 7:40:20. Tuckerman and Lehmer later provided confirmation of this result." Curt Noll and Laura Nickel, ''The 25th and 26th Mersenne Primes'', Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]</ref>
| |
| | [[Landon Curt Noll]] & Laura Nickel
| |
| | LLT / [[CDC Cyber]] 174
| |
| |-
| |
| | style="text-align:right;"| 26
| |
| | style="text-align:right;"| 23,209
| |
| | style="text-align:right;"| 402874115…779264511
| |
| | style="text-align:right;"| 6,987
| |
| | 1979 February 9<ref>"Of the 125 remaining M<sub>p</sub> only M<sub>23209</sub> was found to be prime. The test was completed on February 9, 1979 at 4:06 after 8:39:37 of CPU time. Lehmer and McGrogan later confirmed the result." Curt Noll and Laura Nickel, ''The 25th and 26th Mersenne Primes'', Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]</ref>
| |
| | Landon Curt Noll
| |
| | LLT / CDC Cyber 174
| |
| |-
| |
| | style="text-align:right;"| 27
| |
| | style="text-align:right;"| 44,497
| |
| | style="text-align:right;"| 854509824…011228671
| |
| | style="text-align:right;"| 13,395
| |
| | 1979 April 8<ref>David Slowinski, "Searching for the 27th Mersenne Prime", ''Journal of Recreational Mathematics'', v. 11(4), 1978–79, pp. 258–261, MR 80g #10013</ref><ref>"The 27th Mersenne prime. It has 13395 digits and equals 2<sup>44497</sup>-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> − 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;"| 86,243
| |
| | style="text-align:right;"| 536927995…433438207
| |
| | style="text-align:right;"| 25,962
| |
| | 1982 September 25
| |
| | David Slowinski
| |
| | LLT / Cray 1
| |
| |-
| |
| | style="text-align:right;"| 29
| |
| | style="text-align:right;"| 110,503
| |
| | style="text-align:right;"| 521928313…465515007
| |
| | style="text-align:right;"| 33,265
| |
| | 1988 January 29<ref>"An FFT containing 8192 complex elements, which was the minimum size required to test M<sub>110503</sub>, ran aproximately 11 minutes on the SX-2. The discovery of 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 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;"| 512740276…730061311
| |
| | style="text-align:right;"| 39,751
| |
| | 1983 September 19<ref>"Slowinski, a software engineer for Cray Research Inc. in Chippewa Falls, discovered the number at 11:36 a.m. Monday. [i.e. 1983 September 19]" [http://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, ] [http://news.google.com/newspapers?id=roFQAAAAIBAJ&sjid=VRIEAAAAIBAJ&pg=4379%2C4887110 p. 11] [retrieved 2012-10-23]</ref>
| |
| | David Slowinski
| |
| | LLT / [[Cray X-MP]]
| |
| |-
| |
| | style="text-align:right;"| 31
| |
| | style="text-align:right;"| 216,091
| |
| | style="text-align:right;"| 746093103…815528447
| |
| | style="text-align:right;"| 65,050
| |
| | 1985 September 1<ref>"The number is the 30th known example of a Mersenne prime, a number divisible only by 1 and itself and written in the form 2<sup>p</sup>-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]</ref><ref>"Slowinski's program also found the 28th in 1982, the 29th in 1983, and the 30th [known at that time] this past Labor Day weekend. [i.e. August 31-September 1, 1985]" [http://www.chron.com/CDA/archives/archive.mpl/1985_39205/supercomputer-chevron-calculating-device-finds-a-b.html Rad Sallee, "`Supercomputer'/Chevron calculating device finds a bigger prime number" ''Houston Chronicle'', Friday 09/20/1985, Section 1, Page 26, 4 Star Edition] [retrieved 2012-10-23]</ref>
| |
| | David Slowinski
| |
| | LLT / Cray X-MP/24
| |
| |-
| |
| | style="text-align:right;"| 32
| |
| | style="text-align:right;"| 756,839
| |
| | style="text-align:right;"| 174135906…544677887
| |
| | style="text-align:right;"| 227,832
| |
| | 1992 February 17
| |
| | David Slowinski & [[Paul Gage]]
| |
| | LLT / [[Harwell Lab]]'s [[Cray-2]]<ref>The Prime Pages, [http://primes.utm.edu/notes/756839.html The finding of the 32''nd'' Mersenne].</ref>
| |
| |-
| |
| | style="text-align:right;"| 33
| |
| | style="text-align:right;"| 859,433
| |
| | style="text-align:right;"| 129498125…500142591
| |
| | style="text-align:right;"| 258,716
| |
| | 1994 January 4<ref>Chris Caldwell, [http://www.math.unicaen.fr/~reyssat/largest.html The Largest Known Primes].</ref><ref>[http://www.0x07bell.net/WWWMASTER/CrayWWWStuff/prfoldercomp/PRIME_NUMBER.940110.txt Crays press release]</ref><ref>[https://groups.google.com/forum/#!msg/sci.crypt/XXNkt07IE2s/eDeaWg5-mtIJ Slowinskis email]</ref>
| |
| | David Slowinski & Paul Gage
| |
| | LLT / [[Cray C90]]
| |
| |-
| |
| | style="text-align:right;"| 34
| |
| | style="text-align:right;"| 1,257,787
| |
| | style="text-align:right;"| 412245773…089366527
| |
| | style="text-align:right;"| 378,632
| |
| | 1996 September 3<ref>Silicon Graphics' press release http://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>-1].</ref>
| |
| | LLT / [[Cray T90|Cray T94]]
| |
| |-
| |
| | style="text-align:right;"| 35
| |
| | style="text-align:right;"| 1,398,269
| |
| | style="text-align:right;"| 814717564…451315711
| |
| | style="text-align:right;"| 420,921
| |
| | 1996 November 13
| |
| | [[Great Internet Mersenne Prime Search|GIMPS]] / Joel Armengaud<ref>[http://www.mersenne.org/primes/1398269.htm GIMPS Discovers 35th Mersenne Prime].</ref>
| |
| | LLT / [[Prime95]] on 90 MHz [[Pentium]] PC
| |
| |-
| |
| | style="text-align:right;"| 36
| |
| | style="text-align:right;"| 2,976,221
| |
| | style="text-align:right;"| 623340076…729201151
| |
| | style="text-align:right;"| 895,932
| |
| | 1997 August 24
| |
| | GIMPS / Gordon Spence<ref>[http://www.mersenne.org/primes/2976221.htm GIMPS Discovers 36th Known Mersenne Prime].</ref>
| |
| | LLT / Prime95 on 100 MHz Pentium PC
| |
| |-
| |
| | style="text-align:right;"| 37
| |
| | style="text-align:right;"| 3,021,377
| |
| | style="text-align:right;"| 127411683…024694271
| |
| | style="text-align:right;"| 909,526
| |
| | 1998 January 27
| |
| | GIMPS / Roland Clarkson<ref>[http://www.mersenne.org/primes/3021377.htm GIMPS Discovers 37th Known Mersenne Prime].</ref>
| |
| | LLT / Prime95 on 200 MHz Pentium PC
| |
| |-
| |
| | style="text-align:right;"| 38
| |
| | style="text-align:right;"| 6,972,593
| |
| | style="text-align:right;"| 437075744…924193791
| |
| | style="text-align:right;"| 2,098,960
| |
| | 1999 June 1
| |
| | GIMPS / Nayan Hajratwala<ref>[http://www.mersenne.org/primes/6972593.htm GIMPS Finds First Million-Digit Prime, Stakes Claim to $50,000 EFF Award].</ref>
| |
| | LLT / Prime95 on 350 MHz [[Pentium II]] [[IBM Aptiva]]
| |
| |-
| |
| | style="text-align:right;"| 39
| |
| | style="text-align:right;"| 13,466,917
| |
| | style="text-align:right;"| 924947738…256259071
| |
| | style="text-align:right;"| 4,053,946
| |
| | 2001 November 14
| |
| | GIMPS / Michael Cameron<ref>GIMPS, [http://www.mersenne.org/primes/13466917.htm Researchers Discover Largest Multi-Million-Digit Prime Using Entropia Distributed Computing Grid].</ref>
| |
| | LLT / Prime95 on 800 MHz [[Athlon Thunderbird|Athlon T-Bird]]
| |
| |-
| |
| | style="text-align:right;"| 40
| |
| | style="text-align:right;"| 20,996,011
| |
| | style="text-align:right;"| 125976895…855682047
| |
| | style="text-align:right;"| 6,320,430
| |
| | 2003 November 17
| |
| | GIMPS / Michael Shafer<ref>GIMPS, [http://www.mersenne.org/primes/20996011.htm Mersenne Project Discovers Largest Known Prime Number on World-Wide Volunteer Computer Grid].</ref>
| |
| | LLT / Prime95 on 2 GHz [[Dell Dimension]]
| |
| |-
| |
| | style="text-align:right;"| 41
| |
| | style="text-align:right;"| 24,036,583
| |
| | style="text-align:right;"| 299410429…733969407
| |
| | style="text-align:right;"| 7,235,733
| |
| | 2004 May 15
| |
| | GIMPS / Josh Findley<ref>GIMPS, [http://www.mersenne.org/primes/24036583.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>24,036,583</sup>-1].</ref>
| |
| | LLT / Prime95 on 2.4 GHz [[Pentium 4]] PC
| |
| |-
| |
| | style="text-align:right;"| 42
| |
| | style="text-align:right;"| 25,964,951
| |
| | style="text-align:right;"| 122164630…577077247
| |
| | style="text-align:right;"| 7,816,230
| |
| | 2005 February 18
| |
| | GIMPS / Martin Nowak<ref>GIMPS, [http://www.mersenne.org/primes/25964951.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>25,964,951</sup>-1].</ref>
| |
| | LLT / Prime95 on 2.4 GHz Pentium 4 PC
| |
| |-
| |
| | style="text-align:right;"| 43{{ref label|unverified_index|*|^ *}}
| |
| | style="text-align:right;"| 30,402,457
| |
| | style="text-align:right;"| 315416475…652943871
| |
| | style="text-align:right;"| 9,152,052
| |
| | 2005 December 15
| |
| | GIMPS / [[Curtis Cooper (mathematician)|Curtis Cooper]] & Steven Boone<ref>GIMPS, [http://www.mersenne.org/primes/30402457.htm Mersenne.org Project Discovers New Largest Known Prime Number, 2<sup>30,402,457</sup>-1].</ref>
| |
| | LLT / Prime95 on 2 GHz Pentium 4 PC
| |
| |-
| |
| | style="text-align:right;"| 44{{ref label|unverified_index|*|^ *}}
| |
| | style="text-align:right;"| 32,582,657
| |
| | style="text-align:right;"| 124575026…053967871
| |
| | style="text-align:right;"| 9,808,358
| |
| | 2006 September 4
| |
| | GIMPS / Curtis Cooper & Steven Boone<ref>GIMPS, [http://www.mersenne.org/primes/32582657.htm Mersenne.org Project Discovers Largest Known Prime Number, 2<sup>32,582,657</sup>-1].</ref>
| |
| | LLT / Prime95 on 3 GHz Pentium 4 PC
| |
| |-
| |
| | style="text-align:right;"| 45{{ref label|unverified_index|*|^ *}}
| |
| | style="text-align:right;"| 37,156,667
| |
| | style="text-align:right;"| 202254406…308220927
| |
| | style="text-align:right;"| 11,185,272
| |
| | 2008 September 6
| |
| | GIMPS / Hans-Michael Elvenich<ref name="mp">[http://mersenne.org/primes/m45and46.htm Titanic Primes Raced to Win $100,000 Research Award]. Retrieved on 2008-09-16.</ref>
| |
| | LLT / Prime95 on 2.83 GHz [[Core 2 Duo]] PC
| |
| |-
| |
| | style="text-align:right;"| 46{{ref label|unverified_index|*|^ *}}
| |
| | style="text-align:right;"| 42,643,801
| |
| | style="text-align:right;"| 169873516…562314751
| |
| | style="text-align:right;"| 12,837,064
| |
| | 2009 April 12{{ref label|date_issue|**|^ **}}
| |
| | GIMPS / Odd M. Strindmo<ref>"On April 12th [2009], the 47th known Mersenne prime, 2<sup>42,643,801</sup>-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>
| |
| | LLT / Prime95 on 3 GHz Core 2 PC
| |
| |-
| |
| | style="text-align:right;"| 47{{ref label|unverified_index|*|^ *}}
| |
| | style="text-align:right;"| 43,112,609
| |
| | style="text-align:right;"| 316470269…697152511
| |
| | style="text-align:right;"| 12,978,189
| |
| | 2008 August 23
| |
| | GIMPS / Edson Smith<ref name="mp" />
| |
| | LLT / Prime95 on [[Dell Optiplex]] 745
| |
| |-
| |
| | style="text-align:right;"| 48{{ref label|unknown_index|*|^ *}}
| |
| | style="text-align:right;"| 57,885,161
| |
| | style="text-align:right;"| 581887266…724285951
| |
| | style="text-align:right;"| 17,425,170
| |
| | 2013 January 25
| |
| | GIMPS / Curtis Cooper<ref name="m48" />
| |
| | LLT / Prime95 on 3 GHz Core 2 Duo PC<ref>{{cite web|last=Woltman|first=George|title=NEW MERSENNE PRIME! TOTALLY MERSENNE THIS TIME! thread|url=http://www.mersenneforum.org/showpost.php?p=327763&postcount=372|publisher=''mersenneforum''|accessdate=5 February 2013}}</ref>
| |
| |}
| |
| | |
| {{note label|unverified_index|*|^ *}} <small>It is not verified whether any undiscovered Mersenne primes exist between the 42nd (''M''<sub>25,964,951</sub>) and the 48th (''M''<sub>57,885,161</sub>) on this chart; the ranking is therefore provisional. All Mersenne numbers below the 47th (''M''<sub>43,112,609</sub>) in the interval have been tested at least once but some have not been double-checked. Some Mersenne numbers above the 47th have not yet been tested.<ref name="GIMPS Milestones">[http://www.mersenne.org/report_milestones/ GIMPS Milestones Report]. Retrieved 2013-02-14</ref> Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered ''after'' the 30th and the 31st. Similarly, ''M''<sub>43,112,609</sub> was followed by two smaller Mersenne primes, first 2 weeks later and then 8 months later.</small>
| |
| | |
| {{note label|date_issue|**|^ **}} <small>''M''<sub>42,643,801</sub> was first found by a machine on April 12, 2009; however, no human took notice of this fact until June 4. Thus, either April 12 or June 4 may be considered the 'discovery' date. The discoverer, Strindmo, apparently used the alias Stig M. Valstad.
| |
| </small>
| |
| | |
| To help visualize the size of the 48th known Mersenne prime, it would require 4,647 pages to display the number in base 10 with 75 digits per line and 50 lines per page.
| |
| | |
| The largest known Mersenne prime (2<sup>57,885,161</sup> − 1) is also the [[largest known prime number]].<ref name="m48" /> ''M''<sub>43,112,609</sub> was the first discovered prime number with more than 10 million base-10 digits.
| |
| | |
| In modern times, the largest known prime has almost always been a Mersenne prime.<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>
| |
| | |
| ==Factorization of composite Mersenne numbers==
| |
| The factorization of a prime number is by definition the number itself. This section is about composite numbers. Mersenne numbers are very good test cases for the [[special number field sieve]] algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. {{As of|2012|8}}, 2{{sup|1,061}} − 1 is the record-holder,<ref>[[Greg Childers]], [http://physics.fullerton.edu/gchilders/M1061.pdf "Factorization of a 1061-bit number by the Special Number Field Sieve"].</ref> using the special number field sieve. See [[integer factorization records]] for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a [[primality test]] on the cofactor. {{As of|2014|January}}, the composite Mersenne number with largest proven prime factor is 2{{sup|63,703}} − 1 = {{nobreak|42,808,417 × ''p''}}, where ''p'' has 19,169 digits and was proven prime with [[Elliptic curve primality proving|ECPP]].<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=49 The Top Twenty: Mersenne cofactor] at The [[Prime Pages]]. Retrieved 2014-01-25.</ref> {{As of|2014|January}}, the largest factorization with [[probable prime]] factors allowed is 2{{sup|1,168,183}} − 1 = {{nobreak|54,763,676,838,381,762,583 × ''q''}}, where ''q'' is a 351,639-digit probable prime.<ref>{{cite web|url=http://www.primenumbers.net/prptop/searchform.php?form=(2^p-1)/%3F&action=Search |title=PRP Top Records |author=Henri Lifchitz and Renaud Lifchitz |accessdate=2014-01-28}}</ref>
| |
| | |
| ==Mersenne numbers in nature and elsewhere==
| |
| In computer science, [[signedness|unsigned]] ''n''-bit [[integer (computer science)|integers]] can be used to express numbers up to ''M<sub>n</sub>''. [[Signed number representations|Signed]] (''n'' + 1)-bit integers can express values between −(''M<sub>n</sub>'' + 1) and ''M<sub>n</sub>'', using the [[two's complement]] representation.
| |
| | |
| In the mathematical problem [[Tower of Hanoi]], solving a puzzle with an ''n''-disc tower requires ''M<sub>n</sub>'' steps, assuming no mistakes are made.<ref>{{cite book|last=Petković|first=Miodrag|title=Famous Puzzles of Great Mathematicians|year=2009|publisher=AMS Bookstore|isbn=0-8218-4814-3|pages=197}}</ref>
| |
| | |
| The [[asteroid]] with [[minor planet]] number 8191 is named [[8191 Mersenne]] after Marin Mersenne, because 8191 is a Mersenne prime ([[3 Juno]], [[7 Iris]], [[31 Euphrosyne]] and [[127 Johanna]] having been discovered and named during the 19th century).<ref>{{cite web|author=Alan Chamberlin |url=http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=8191+Mersenne |title=JPL Small-Body Database Browser |publisher=Ssd.jpl.nasa.gov |date= |accessdate=2011-05-21}}</ref>
| |
| | |
| ==See also==
| |
| <div style="-moz-column-count:2; column-count:2;">
| |
| * [[Repunit]]
| |
| * [[Fermat prime]]
| |
| * [[Erdős–Borwein constant]]
| |
| * [[Mersenne conjectures]]
| |
| * [[Mersenne Twister]]
| |
| * [[Prime95]] / [[MPrime]]
| |
| * [[Great Internet Mersenne Prime Search]] (GIMPS)
| |
| * [[Largest known prime number]]
| |
| * [[Double Mersenne number]]
| |
| * [[Wieferich prime]]
| |
| * [[Wagstaff prime]]
| |
| * [[Solinas prime]]
| |
| * [[Gillies' conjecture]]
| |
| </div>
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| ==References==
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| {{reflist|colwidth=30em}}
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| ==External links==
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| {{Wiktionary|Mersenne prime}}
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| {{Wikinewspar2|Record size 17.4 million-digit prime found|Two largest known prime numbers discovered just two weeks apart, one qualifies for $100k prize|Distributed computing discovers largest known prime number|CMSU computing team discovers another record size prime}}
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| *{{springer|title=Mersenne number|id=p/m063480}}
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| *[http://www.mersenne.org GIMPS home page]
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| *[http://v5www.mersenne.org/report_milestones/ GIMPS status] — status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of primes 42–47
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| *''M''<sub>''q''</sub> = (8''x'')<sup>2</sup> − (3''qy'')<sup>2</sup> [http://tony.reix.free.fr/Mersenne/Mersenne8x3qy.pdf Mersenne proof] (PDF)
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| *''M''<sub>''q''</sub> = ''x''<sup>2</sup> + ''d''·''y''<sup>2</sup> [http://www.math.leidenuniv.nl/scripties/jansen.ps math thesis] (PS)
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| *{{cite web|last=Grime|first=James|title=31 and Mersenne Primes|url=http://www.numberphile.com/videos/31.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| *[http://www.utm.edu/research/primes/mersenne/LukeMirror/biblio.htm Mersenne prime bibliography] with hyperlinks to original publications
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| *[http://www.taz.de/pt/2005/03/11/a0355.nf/text report about Mersenne primes] — detection in detail {{de icon}}
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| *[http://mersennewiki.org/index.php/Main_Page GIMPS wiki]
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| *[http://www.garlic.com/~wedgingt/mersenne.html Will Edgington's Mersenne Page] — contains factors for small Mersenne numbers
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| *a [ftp://mersenne.org/gimps/factors.zip file] containing the smallest known factors of many tested Mersenne numbers (requires [ftp://mersenne.org/gimps/decomp.zip program] to open)
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| *[http://www.isthe.com/chongo/tech/math/prime/mersenne.html Decimal digits and English names of Mersenne primes]
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| *[http://primes.utm.edu/curios/page.php/2305843009213693951.html Prime curios: 2305843009213693951]
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| ===MathWorld links===
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| * {{mathworld|urlname=MersenneNumber|title=Mersenne number}}
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| * {{mathworld|urlname=MersennePrime|title=Mersenne prime}}
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| * [http://mathworld.wolfram.com/news/2009-06-07/mersenne-47/ 47th Mersenne Prime Found]
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| {{Prime number classes|state=collapsed}}
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Mersenne Prime}}
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| [[Category:Articles containing proofs]]
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| [[Category:Classes of prime numbers]]
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| [[Category:Unsolved problems in mathematics]]
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| [[Category:Integer sequences]]
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