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{{short description|Prime number of the form (2^n)-1}}
{{Infobox integer sequence
| named_after        = [[Marin Mersenne]]
| publication_year   = 1536<ref>{{Cite book | last = Regius | first = Hudalricus | title = Utrisque Arithmetices Epitome | url = https://books.google.com/?id=hs85AAAAcAAJ&printsec=frontcover&dq=Utriusque+Arithmetices+epitome#v=onepage&q=2047&f=false| year = 1536 }}</ref>
| author             = Regius, H.
| terms_number       = 51
| con_number         = Infinite
| parentsequence     = Mersenne numbers
| first_terms        = [[3 (number)|3]], [[7 (number)|7]], [[31 (number)|31]], [[127 (number)|127]]
| largest_known_term = {{nowrap|2<sup>82,589,933</sup> &minus; 1}} (December 7, 2018)
| OEIS               = A000668
| OEIS_name          = Mersenne primes (of form 2^p - 1 where p is a prime)
}}
In [[mathematics]], a '''Mersenne prime''' is a [[prime number]] that is one less than a [[power of two]].  That is, it is a prime number of the form {{math|''M<sub>n</sub>'' {{=}} 2<sup>''n''</sup> − 1}} for some [[integer]] {{math|''n''}}.  They are named after [[Marin Mersenne]], a French [[Minim (religious order)|Minim friar]], who studied them in the early 17th century.

The [[exponentiation|exponents]] {{math|''n''}} which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31,&nbsp;... {{OEIS|id=A000043}} and the resulting Mersenne primes are [[3 (number)|3]], [[7 (number)|7]], [[31 (number)|31]], [[127 (number)|127]], 8191, 131071, 524287, [[2147483647]],&nbsp;... {{OEIS|id=A000668}}.

If {{math|''n''}} is a [[composite number]] then so is {{math|2<sup>''n''</sup> − 1}}. ({{math|2<sup>''ab''</sup> − 1}} is divisible by both {{math|2<sup>''a''</sup> − 1}} and {{math|2<sup>''b''</sup> − 1}}.) This definition is therefore equivalent to the definition as a prime number of the form {{math|''M<sub>p</sub>'' {{=}} 2<sup>''p''</sup> − 1}} for some prime {{math|''p''}}.

More generally, numbers of the form {{math|''M<sub>n</sub>'' {{=}} 2<sup>''n''</sup> − 1}} without the primality requirement may be called '''Mersenne numbers'''.  Sometimes, however, Mersenne numbers are defined to have the additional requirement that {{math|''n''}} be prime.
The smallest composite Mersenne number with prime exponent ''n'' is {{nowrap|2<sup>11</sup> − 1 {{=}} 2047 {{=}} 23 × 89}}.

Mersenne primes {{math|''M''<sub>''p''</sub>}} are also noteworthy due to their [[#Perfect numbers|connection to perfect numbers]].

{{As of|2019|6|url=http://primes.utm.edu/top20/page.php?id=4}}, 51 Mersenne primes are known. The [[largest known prime number]], {{nowrap|2<sup>82,589,933</sup> − 1}}, is a Mersenne prime.<ref name="GIMPS-2018">{{cite web |title=GIMPS Project Discovers Largest Known Prime Number: 2<sup>82,589,933</sup>-1 |url=https://www.mersenne.org/primes/press/M82589933.html |date=21 December 2018 |work=Mersenne Research, Inc. |accessdate=21 December 2018 }}</ref> 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.

==About Mersenne primes==
{{unsolved|mathematics|Are there infinitely many Mersenne primes?}}
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]]). For these primes {{math|''p''}}, {{math|2''p'' + 1}} (which is also prime) will divide {{math|''M<sub>p</sub>''}}, for example, {{math|23&nbsp;{{!}}&nbsp;''M''<sub>11</sub>}}, {{math|47&nbsp;{{!}}&nbsp;''M''<sub>23</sub>}}, {{math|167&nbsp;{{!}}&nbsp;''M''<sub>83</sub>}}, {{math|263&nbsp;{{!}}&nbsp;''M''<sub>131</sub>}}, {{math|359&nbsp;{{!}}&nbsp;''M''<sub>179</sub>}}, {{math|383&nbsp;{{!}}&nbsp;''M''<sub>191</sub>}}, {{math|479&nbsp;{{!}}&nbsp;''M''<sub>239</sub>}}, and {{math|503&nbsp;{{!}}&nbsp;''M''<sub>251</sub>}} {{OEIS|id=A002515}}. Since for these primes {{math|''p''}}, {{math|2''p'' + 1}} is congruent to 7 mod 8, so 2 is a [[quadratic residue]] mod {{math|2''p'' + 1}}, and the [[multiplicative order]] of 2 mod {{math|2''p'' + 1}} must divide <math>\frac{(2p+1)-1}{2}</math> = {{math|''p''}}. Since {{math|''p''}} is a prime, it must be {{math|''p''}} or 1. However, it cannot be 1 since <math>\Phi_1(2)=1</math> and 1 has no [[prime factor]]s, so it must be {{math|''p''}}. Hence, {{math|2''p'' + 1}} divides <math>\Phi_p(2)=2^p-1</math> and <math>2^p-1=M_p</math> cannot be prime.

The first four Mersenne primes are {{math|''M''<sub>2</sub> {{=}} 3}}, {{math|''M''<sub>3</sub> {{=}} 7}}, {{math|''M''<sub>5</sub> {{=}} 31}} and {{math|''M''<sub>7</sub> {{=}} 127}} and because the first Mersenne prime starts at {{math|''M''<sub>2</sub>}}, all Mersenne primes are congruent to 3 (mod 4). Other than {{math|''M''<sub>0</sub> {{=}} 0}} and {{math|''M''<sub>1</sub> {{=}} 1}}, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the [[prime factorization]] of a Mersenne number (&nbsp;{{math|&ge;&nbsp;''M''<sub>2</sub>}}&nbsp;) there must be at least one prime factor congruent to 3 (mod 4).

A basic [[theorem]] about Mersenne numbers states that if {{math|''M''<sub>''p''</sub>}} is prime, then the exponent {{math|''p''}} must also be prime. This follows from the identity

:<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>
This rules out primality for Mersenne numbers with composite exponent, such as {{math|''M''<sub>4</sub> {{=}} 2<sup>4</sup> − 1 {{=}} 15 {{=}} 3 × 5 {{=}} (2<sup>2</sup> − 1) × (1 + 2<sup>2</sup>)}}.

Though the above examples might suggest that {{math|''M<sub>p</sub>''}} is prime for all primes {{math|''p''}}, this is not the case, and the smallest counterexample is the Mersenne number

: {{math|''M''<sub>11</sub> {{=}} 2<sup>11</sup> − 1 {{=}} 2047 {{=}} 23 × 89}}.

The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size.<ref name="Wagstaff" /> Nonetheless, prime values of {{math|''M''<sub>''p''</sub>}} appear to grow increasingly sparse as {{math|''p''}} increases. For example, eight of the first 11 primes {{mvar|p}} give rise to a Mersenne prime {{math|''M''<sub>''p''</sub>}} (the correct terms on Mersenne's original list), while {{math|''M''<sub>''p''</sub>}} is prime for only 43 of the first two million prime numbers (up to 32,452,843).

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, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. 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]].

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. They are also used in [[primitive polynomial (field theory)#Primitive trinomials|primitive trinomials]], which can also be used to create PRNGs with very large periods.

==Perfect numbers==
{{main|Euclid–Euler theorem}}
Mersenne primes {{math|''M''<sub>''p''</sub>}} are also noteworthy due to their connection with [[perfect number]]s. In the 4th century BC, [[Euclid]] proved that if {{math|2<sup>''p''</sup> − 1}} is prime, then {{math|2<sup>''p'' − 1</sup>(2<sup>''p''</sup> − 1}}) is a perfect number. This number, also expressible as {{math|{{sfrac|''M''<sub>''p''</sub>(''M<sub>p</sub>'' + 1)|2}}}}, is the {{math|''M<sub>p</sub>''}}th [[triangular number]] and the {{math|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> This is known as the [[Euclid–Euler theorem]]. It is unknown whether there are any [[odd perfect number]]s.

==History==
{| class="wikitable infobox" style="width:1px;padding-left:1em;"
! style="color:#ff0000;background:#00ffff"|2
! style="color:#ff0000;background:#00ffff"|3
! style="color:#ff0000;background:#00ffff"|5
! style="color:#ff0000;background:#00ffff"|7
| style="text-align: center;"|11
! style="color:#ff0000;background:#00ffff"|13
! style="color:#ff0000;background:#00ffff"|17
! style="color:#ff0000;background:#00ffff"|19
|-
| style="text-align: center;"|23
| style="text-align: center;"|29
! style="color:#ff0000;background:#00ffff"|31
| style="text-align: center;"|37
| style="text-align: center;"|41
| style="text-align: center;"|43
| style="text-align: center;"|47
| style="text-align: center;"|53
|-
| style="text-align: center;"|59
! style="background:#00ffff"|61
! style="color:#ff0000;|67
| style="text-align: center;"|71
| style="text-align: center;"|73
| style="text-align: center;"|79
| style="text-align: center;"|83
! style="background:#00ffff"|89
|-
| style="text-align: center;"|97
| style="text-align: center;"|101
| style="text-align: center;"|103
! style="background:#00ffff"|107
| style="text-align: center;"|109
| style="text-align: center;"|113
! style="color:#ff0000;background:#00ffff"|127
| style="text-align: center;"|131
|-
| style="text-align: center;"|137
| style="text-align: center;"|139
| style="text-align: center;"|149
| style="text-align: center;"|151
| style="text-align: center;"|157
| style="text-align: center;"|163
| style="text-align: center;"|167
| style="text-align: center;"|173
|-
| style="text-align: center;"|179
| style="text-align: center;"|181
| style="text-align: center;"|191
| style="text-align: center;"|193
| style="text-align: center;"|197
| style="text-align: center;"|199
| style="text-align: center;"|211
| style="text-align: center;"|223
|-
| style="text-align: center;"|227
| style="text-align: center;"|229
| style="text-align: center;"|233
| style="text-align: center;"|239
| style="text-align: center;"|241
| style="text-align: center;"|251
! style="color:#ff0000;|257
| style="text-align: center;"|263
|-
| style="text-align: center;"|269
| style="text-align: center;"|271
| style="text-align: center;"|277
| style="text-align: center;"|281
| style="text-align: center;"|283
| style="text-align: center;"|293
| style="text-align: center;"|307
| style="text-align: center;"|311
|-
| colspan="8"|The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold.
|}

Mersenne primes take their name from the 17th-century [[French people|French]] scholar [[Marin Mersenne]], who compiled what was supposed to be a list of Mersenne primes with exponents up to 257.  The exponents listed by Mersenne were as follows:

::2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.

His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included {{math|''M''<sub>67</sub>}} and {{math|''M''<sub>257</sub>}} (which are composite) and omitted {{math|''M''<sub>61</sub>}}, {{math|''M''<sub>89</sub>}}, and {{math|''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>

[[Édouard Lucas]] proved in 1876 that {{math|''M''<sub>127</sub>}} is indeed prime, as Mersenne claimed.  This was the largest known prime number for 75 years, and the largest ever found by hand.  {{math|''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 {{math|''M''<sub>67</sub>}} is actually composite. No factor was found until a famous talk by [[Frank Nelson Cole]] in 1903.<ref>{{citation|last=Cole|first= F. N.
|title=On the factoring of large numbers|jfm= 34.0216.04|journal=Bull. Amer. Math. Soc. |year=1903|volume=10|issue= 3
|pages=134–137|doi=10.1090/S0002-9904-1903-01079-9}}</ref>  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 {{nowrap|193,707,721 × 761,838,257,287}} and got the same number, then returned to his seat (to applause) without speaking.<ref>{{cite book
| title     = Mathematics, queen and servant of science
| url     = https://archive.org/details/mathematicsqueen0000bell_n9l1
| url-access = registration
| author    = Bell, E.T. and Mathematical Association of America
| year      = 1951
| publisher = McGraw-Hill New York}} p. 228.</ref> He later said that the result had taken him "three years of Sundays" to find.<ref>{{cite news|url=http://news.bbc.co.uk/dna/place-lancashire/plain/A670051 |title=h2g2: Mersenne Numbers |work=BBC News |url-status=dead |archiveurl=https://web.archive.org/web/20141205081512/http://news.bbc.co.uk/dna/place-lancashire/plain/A670051 |archivedate=December 5, 2014 }}</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.

==Searching for Mersenne primes==
Fast algorithms for finding Mersenne primes are available, and {{as of|2019|6|lc=y}} the seven [[largest known prime number]]s are Mersenne primes.

The first four Mersenne primes {{math|''M''<sub>2</sub> {{=}} 3}}, {{math|''M''<sub>3</sub> {{=}} 7}}, {{math|''M''<sub>5</sub> {{=}} 31}} and {{math|''M''<sub>7</sub> {{=}} 127}} were known in antiquity. The fifth, {{math|''M''<sub>13</sub> {{=}} 8191}}, was discovered anonymously before 1461; the next two ({{math|''M''<sub>17</sub>}} and {{math|''M''<sub>19</sub>}}) were found by [[Pietro Cataldi]] in 1588. After nearly two centuries, {{math|''M''<sub>31</sub>}} was verified to be prime by [[Leonhard Euler]] in 1772. The next (in historical, not numerical order) was {{math|''M''<sub>127</sub>}}, found by [[Édouard Lucas]] in 1876, then {{math|''M''<sub>61</sub>}} by [[Ivan Mikheevich Pervushin]] in 1883. Two more ({{math|''M''<sub>89</sub>}} and {{math|''M''<sub>107</sub>}}) were found early in the 20th century, by [[R. E. Powers]] in 1911 and 1914, respectively.

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 {{math|''p'' > 2}}, {{math|''M<sub>p</sub>'' {{=}} 2<sup>''p''</sup> − 1}} is prime [[if and only if]] {{math|''M<sub>p</sub>''}} divides {{math|''S''<sub>''p'' − 2</sub>}}, where {{math|''S''<sub>0</sub> {{=}} 4}} and {{math|''S<sub>k</sub>'' {{=}} (''S''<sub>''k'' − 1</sub>)<sup>2</sup> − 2}} for {{math|''k'' > 0}}.

During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229.<ref>{{cite journal |journal=Scripta Mathematica |volume= 18 |year=1952 |pp=122–131 |title=A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes |author=Horace S. Uhler |url=https://primes.utm.edu/mersenne/LukeMirror/lit/lit_019.htm}}</ref> Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes; in relative terms: the next Mersenne prime exponent, 521, would turn out to be more than four times larger than the previous record of 127.

[[File: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 doubly logarithmic in the value of the prime.]]
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, {{math|''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, {{math|''M''<sub>607</sub>}}, was found by the computer a little less than two hours later. Three more&nbsp;— {{math|''M''<sub>1279</sub>}}, {{math|''M''<sub>2203</sub>}}, {{math|''M''<sub>2281</sub>}}&nbsp;— were found by the same program in the next several months. {{math|''M''<sub>4253</sub>}} is the first Mersenne prime that is [[titanic prime|titanic]], {{math|''M''<sub>44,497</sub>}} is the first [[gigantic prime|gigantic]], and {{math|''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. The number of digits in the decimal representation of {{math|''M<sub>n</sub>''}} equals {{math|⌊''n'' × log<sub>10</sub>2⌋ + 1}}, where {{math|⌊''x''⌋}} denotes the [[floor function]] (or equivalently {{math|⌊log<sub>10</sub>''M<sub>n</sub>''⌋ + 1}}).

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 was the eighth Mersenne prime discovered at UCLA.<ref>{{cite news|url=http://www.latimes.com/news/science/la-sci-prime27-2008sep27,0,2746766.story |title=UCLA mathematicians discover a 13-million-digit prime number|newspaper=Los Angeles Times|date=2008-09-27 |accessdate=2011-05-21|first=Thomas H.|last=Maugh II}}</ref>

On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is {{nowrap|2<sup>42,643,801</sup> − 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.

On January 25, 2013, [[Curtis Cooper (mathematician)|Curtis Cooper]], a mathematician at the [[University of Central Missouri]], discovered a 48th Mersenne prime, {{nowrap|2<sup>57,885,161</sup> − 1}} (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.<ref>{{cite magazine| url=http://www.scientificamerican.com/article.cfm?id=largest-prime-number-disc |title=Largest Prime Number Discovered |author=Tia Ghose |magazine=[[Scientific American]] |accessdate=2013-02-07}}</ref>

On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, {{nowrap|2<sup>74,207,281</sup> − 1}} (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network.<ref name="GIMPS-2016">{{cite web |last=Cooper |first=Curtis |title=Mersenne Prime Number discovery – 2<sup>74207281</sup>&nbsp;−&nbsp;1 is Prime! |url=http://www.mersenne.org/primes/?press=M74207281 |date=7 January 2016 |work=Mersenne Research, Inc. |accessdate=22 January 2016 }}</ref><ref>{{cite news|url=https://www.newscientist.com/article/2073909-prime-number-with-22-million-digits-is-the-biggest-ever-found/|title=Prime number with 22 million digits is the biggest ever found|last=Brook|first=Robert|date=January 19, 2016|work=[[New Scientist]]|accessdate=19 January 2016}}</ref><ref name="NYT-20160121">{{cite news |last=Chang |first=Kenneth |title=New Biggest Prime Number = 2 to the 74 Mil ... Uh, It’s Big |url=https://www.nytimes.com/2016/01/22/science/new-biggest-prime-number-mersenne-primes.html |date=21 January 2016 |work=[[The New York Times]] |accessdate=22 January 2016 }}</ref> This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years.

On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below M<sub>37,156,667</sub>, thus officially confirming its position as the 45th Mersenne prime.<ref>{{cite web|url = https://web.archive.org/web/20160903040753/http://www.mersenne.org/report_milestones/|title = Milestones}}</ref>

On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in Germantown, Tennessee, had found a 50th Mersenne prime, {{nowrap|2<sup>77,232,917</sup> − 1}} (a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network.<ref>{{Cite web|url=https://www.mersenne.org/primes/?press=M77232917|title=Mersenne Prime Discovery - 2^77232917-1 is Prime!|website=www.mersenne.org|language=en|access-date=2018-01-03}}</ref>

On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number, {{nowrap|2<sup>82,589,933</sup> − 1}}, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.<ref>{{Cite web|url=https://www.mersenne.org/primes/?press=M82589933|title=GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1|language=en|access-date=2019-01-01}}</ref>

==Theorems about Mersenne numbers==
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# If {{math|''a''}} and {{math|''p''}} are natural numbers such that {{math|''a<sup>p</sup>'' − 1}} is prime, then {{math|''a'' {{=}} 2}} or {{math|''p'' {{=}} 1}}.
#* '''Proof''': {{math|''a'' ≡ 1 ([[Modular arithmetic|mod]] ''a'' − 1)}}. Then {{math|''a<sup>p</sup>'' ≡ 1 (mod ''a'' − 1)}}, so {{math|''a<sup>p</sup>'' − 1 ≡ 0 (mod ''a'' − 1)}}. Thus {{math|''a'' − 1 {{!}} ''a<sup>p</sup>'' − 1}}. However, {{math|''a<sup>p</sup>'' − 1}} is prime, so {{math|''a'' − 1 {{=}} ''a<sup>p</sup>'' − 1}} or {{math|''a'' − 1 {{=}} ±1}}. In the former case, {{math|''a'' {{=}} ''a<sup>p</sup>''}}, hence {{math|''a'' {{=}} 0,1}} (which is a contradiction, as neither −1 nor 0 is prime) or {{math|''p'' {{=}} 1.}} In the latter case, {{math|''a'' {{=}} 2}} or {{math|''a'' {{=}} 0}}. If {{math|''a'' {{=}} 0}}, however, {{math|0<sup>''p''</sup> − 1 {{=}} 0 − 1 {{=}} −1}} which is not prime. Therefore, {{math|''a'' {{=}} 2}}.
# If {{math|2<sup>''p''</sup> − 1}} is prime, then {{math|''p''}} is prime.
#* '''Proof''': suppose that {{math|''p''}} is composite, hence can be written {{math|''p'' {{=}} ''ab''}} with ''a'' and {{nowrap|''b'' > 1}}. Then {{math|2<sup>''p''</sup> − 1}} {{math|{{=}} 2<sup>''ab''</sup> − 1}} {{math|{{=}} (2<sup>''a''</sup>)<sup>''b''</sup> − 1}} {{math|{{=}} (2<sup>''a''</sup> − 1)<big>(</big>(2<sup>''a''</sup>)<sup>''b'' − 1</sup> + (2<sup>''a''</sup>)<sup>''b'' − 2</sup> + … + 2<sup>''a''</sup> + 1<big>)</big>}} so {{math|2<sup>''p''</sup> − 1}} is composite. By contrapositive, if {{math|2<sup>''p''</sup> − 1}} is prime then ''p'' is prime.
# If {{math|''p''}} is an odd prime, then every prime {{math|''q''}} that divides {{math|2<sup>''p''</sup> − 1}} must be 1 plus a multiple of {{math|2''p''}}. This holds even when {{math|2<sup>''p''</sup> − 1}} is prime.
#* For example, {{nowrap|2<sup>5</sup> − 1 {{=}} 31}} is prime, and {{nowrap|31 {{=}} 1 + 3 × (2 × 5)}}.  A composite example is {{nowrap|2<sup>11</sup> − 1 {{=}} 23 × 89}}, where {{nowrap|23 {{=}} 1 + (2 × 11)}} and {{nowrap|89 {{=}} 1 + 4 × (2 × 11)}}.
#* '''Proof''': By [[Fermat's little theorem]], {{math|''q''}} is a factor of {{math|2<sup>''q'' − 1</sup> − 1}}. Since {{math|''q''}} is a factor of {{math|2<sup>''p''</sup> − 1}}, for all positive integers {{math|''c''}}, {{math|''q''}} is also a factor of {{math|2<sup>''pc''</sup> − 1}}. Since {{math|''p''}} is prime and {{math|''q''}} is not a factor of {{nowrap|2<sup>1</sup> − 1}}, {{math|''p''}} is also the smallest positive integer {{math|''x''}} such that {{math|''q''}} is a factor of {{math|2<sup>''x''</sup> − 1}}. As a result, for all positive integers {{math|''x''}}, {{math|''q''}} is a factor of {{math|2<sup>''x''</sup> − 1}} if and only if {{math|''p''}} is a factor of {{math|''x''}}. Therefore, since {{math|''q''}} is a factor of {{math|2<sup>''q'' − 1</sup> − 1}}, {{math|''p''}} is a factor of {{math|''q'' − 1}} so {{math|''q'' ≡ 1 (mod ''p'')}}. Furthermore, since {{math|''q''}} is a factor of {{math|2<sup>''p''</sup> − 1}}, which is odd, {{math|''q''}} is odd. Therefore, {{math|''q'' ≡ 1 (mod 2''p'')}}.
#* This fact leads to a proof of [[Euclid's theorem]], which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime {{math|''p''}}, all primes dividing {{math|2<sup>''p''</sup> − 1}} are larger than {{math|''p''}}; thus there are always larger primes than any particular prime.
#* It follows from this fact that for every prime p > 2, there is at least one prime of the form 2kp+1 less than or equal to M<sub>p</sub>, for some integer k.
# If {{math|''p''}} is an odd prime, then every prime {{math|''q''}} that divides {{math|2<sup>''p''</sup> − 1}} is congruent to {{nowrap|±1 (mod 8)}}.
#* '''Proof''': {{math|2<sup>''p'' + 1</sup> ≡ 2 (mod ''q'')}}, so {{math|2<sup>{{sfrac|1|2}}(p + 1)</sup>}} is a square root of {{math|2 mod ''q''}}.  By [[quadratic reciprocity]], every prime modulo in which the number 2 has a square root is congruent to {{nowrap|±1 (mod 8)}}.
# A Mersenne prime cannot be a [[Wieferich prime]].
#* '''Proof''': We show if {{math|''p'' {{=}} 2<sup>''m''</sup> − 1}} is a Mersenne prime, then the congruence {{math|2<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)}} does not hold. By Fermat's little theorem, {{math|''m'' {{!}} ''p'' − 1}}.  Therefore, one can write {{math|''p'' − 1 {{=}} ''mλ''}}. If the given congruence is satisfied, then {{math|''p''<sup>2</sup> {{!}} 2<sup>''mλ''</sup> − 1}}, therefore {{math|0 ≡ {{sfrac|2<sup>''mλ''</sup> − 1|2<sup>''m''</sup> − 1}}}} {{math|{{=}} 1 + 2<sup>''m''</sup> + 2<sup>2''m''</sup> + ... + 2<sup>(''λ'' − 1)''m''</sup>}} {{math|≡ −''λ'' mod (2<sup>''m''</sup> − 1)}}. Hence {{math|2<sup>''m''</sup> − 1 {{!}} ''λ''}}, and therefore {{math|''λ'' ≥ 2<sup>''m''</sup> − 1}}. This leads to {{math|''p'' − 1  ≥ ''m''(2<sup>''m''</sup> − 1)}}, which is impossible since {{math|''m'' ≥ 2}}.
#If {{math|''m''}} and {{math|''n''}} are natural numbers then {{math|''m''}} and {{math|''n''}} are [[coprime]] if and only if {{math|2<sup>''m''</sup> − 1}} and {{math|2<sup>''n''</sup> − 1}} are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number,<ref>[http://www.garlic.com/~wedgingt/mersenne.html Will Edgington's Mersenne Page] {{webarchive|url=https://web.archive.org/web/20141014102940/http://www.garlic.com/~wedgingt/mersenne.html |date=2014-10-14 }}</ref> so in other words the set of [[pernicious number|pernicious]] Mersenne numbers is pairwise coprime.
# If {{math|''p''}} and {{math|2''p'' + 1}} are both prime (meaning that {{math|''p''}} is a [[Sophie Germain prime]]), and {{math|''p''}} is [[Congruence relation|congruent]] to {{nowrap|3 (mod 4)}}, then {{math|2''p'' + 1}} divides {{math|2<sup>''p''</sup> − 1}}.<ref>{{cite web|url=http://primes.utm.edu/notes/proofs/MerDiv2.html|title=Proof of a result of Euler and Lagrange on Mersenne Divisors|first=Chris K.|last=Caldwell|work=[[Prime Pages]]}}</ref>
#* '''Example''': 11 and 23 are both prime, and {{nowrap|11 {{=}} 2 × 4 + 3}}, so 23 divides {{nowrap|2<sup>11</sup> − 1}}.
#* '''Proof''': Let {{math|''q''}} be {{math|2''p'' + 1}}. By Fermat's little theorem, {{math|2<sup>2''p''</sup> ≡ 1 (mod ''q'')}}, so either {{math|2<sup>''p''</sup> ≡ 1 (mod ''q'')}} or {{math|2<sup>''p''</sup> ≡ −1 (mod ''q'')}}. Supposing latter true, then {{math|2<sup>''p'' + 1</sup> {{=}} (2<sup>{{sfrac|1|2}}(''p'' + 1)</sup>)<sup>2</sup> ≡ −2 (mod ''q'')}}, so −2 would be a quadratic residue mod {{math|''q''}}. However, since {{math|''p''}} is congruent to {{nowrap|3 (mod 4)}}, {{math|''q''}} is congruent to {{nowrap|7 (mod 8)}} and therefore 2 is a quadratic residue mod {{math|''q''}}. Also since {{math|''q''}} is congruent to {{nowrap|3 (mod 4)}}, −1 is a quadratic nonresidue mod {{math|''q''}}, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and {{math|2''p'' + 1}} divides {{math|''M<sub>p</sub>''}}.
# All composite divisors of prime-exponent Mersenne numbers are [[strong pseudoprime]]s to the base 2.
# With the exception of 1, a Mersenne number cannot be a perfect power. In other words, and in accordance with [[Catalan's conjecture|Mihăilescu's theorem]], the equation 2<sup>m</sup>-1 = n<sup>k</sup> has no solutions where m, n, and k are integers with m > 1 and k > 1.

==List of known Mersenne primes==
The table below lists all known Mersenne primes (sequence {{OEIS link|id=A000043}} ({{math|''p''}}) and {{OEIS link|id=A000668}} ({{math|''M<sub>p</sub>''}}) in [[OEIS]]):

{| class="wikitable"
|-
! #
! {{math|''p''}}
! {{math|''M<sub>p</sub>''}}
! {{math|''M<sub>p</sub>''}} digits
! Discovered
! Discoverer
! Method used
|-
| style="text-align:right;"| 1
| style="text-align:right;"| 2
| style="text-align:right;"| [[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]]ians 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. "The Egyptians used ($) in the table above for the first primes {{math|''r''}} = 3, 5, 7, or 11 (also for {{math|''r''}} = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) 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/{{math|''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>&nbsp;−&nbsp;1 and 2<sup>3</sup>&nbsp;−&nbsp;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 [that is, 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." [https://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 sense] to contain an equal amount of prime and incomposite numbers, and secondary and composite numbers." [https://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 [https://books.google.com/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 Arithmetic, 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 have no proof of any knowledge of prime numbers.</ref>
|
|-
| style="text-align:right;"| 2
| style="text-align:right;"| 3
| style="text-align:right;"| [[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">{{cite web|url=http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html|title=Euclid's Elements, Book IX, Proposition 36|publisher=}}</ref>
|
|-
| style="text-align:right;"| 4
| style="text-align:right;"| 7
| style="text-align:right;"| [[127 (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), 1400–1465 (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 [https://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;"| [[2,147,483,647|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&nbsp;+&nbsp;101&nbsp;+&nbsp;102&nbsp;+&nbsp;103&nbsp;+&nbsp;...&nbsp;+&nbsp;10T&nbsp;=&nbsp;S].'' Retrieved 2011-10-02.</ref><ref>http://primes.utm.edu/notes/by_year.html#31 The date and year of discovery is unsure. Dates between 1752 and 1772 are possible.</ref>
| Trial division with modular restrictions<ref>{{cite web|author=Chris K. Caldwell |url=http://primes.utm.edu/notes/proofs/MerDiv.html |title=Modular restrictions on Mersenne divisors |publisher=Primes.utm.edu |date= |accessdate=2011-05-21}}</ref>
|-
| style="text-align:right;"| 9
| style="text-align:right;"| 61
| style="text-align:right;"| 2305843009213693951
| style="text-align:right;"| 19
| 1883 November<ref>“En novembre de l’année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l’assertion que le nombre

{{nowrap|2<sup>61</sup> − 1 {{=}} 2305843009213693951}}

est un nombre premier. /…/ Le tome XLVIII des Mémoires Russes de l’Académie /…/ contient le compte-rendu de la séance du 20 décembre 1883, dans lequel l’objet de la communication du père Pervouchine est indiqué avec précision.” Bulletin de l'Académie Impériale des Sciences de St.-Pétersbourg, s. 3, v. 31, 1887, cols. 532–533. https://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|Ivan M. Pervushin]]
| [[Lucas sequence]]s
|-
| style="text-align:right;"| 10
| style="text-align:right;"| 89
| style="text-align:right;"| 618970019642...137449562111
| style="text-align:right;"| 27
| 1911 June<ref>{{cite journal|jstor=2972574|title=The Tenth Perfect Number|first=R. E.|last=Powers|date=1 January 1911|journal=The American Mathematical Monthly|volume=18|issue=11|pages=195–197|via=|doi=10.2307/2972574}}</ref>
| [[Ralph Ernest Powers]]
| Lucas sequences
|-
| style="text-align:right;"| 11
| style="text-align:right;"| 107
| style="text-align:right;"| 162259276829...578010288127
| style="text-align:right;"| 33
| 1914 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 {{math|''M''<sub>107</sub>}}. Sur ma demande, ces deux auteurs m’ont adressé leurs remarquables résultats, et je m’empresse de les publier dans nos colonnes, avec nos felicitations." p. 103, André Gérardin, ''Nombres de Mersenne'' pp. 85, 103–108 in Sphinx-Œdipe. [Journal mensuel de la curiosité, de concours & de mathématiques.] v. 9, No. 1, 1914.</ref><ref>"Power's cable announcing this same result was sent to the London Math. So. on 1 June 1914." Mersenne's Numbers, ''Scripta Mathematica'', v. 3, 1935, pp. 112–119 http://primes.utm.edu/mersenne/LukeMirror/lit/lit_008s.htm [retrieved 2012-10-13]</ref><ref>http://plms.oxfordjournals.org/content/s2-13/1/1.1.full.pdf Proceedings / London Mathematical Society (1914) s2–13 (1): 1. Result presented at a meeting with London Mathematical Society on June 11, 1914. Retrieved 2011-10-02.</ref>
| Ralph Ernest Powers<ref>The Prime Pages, [http://primes.utm.edu/notes/fauquem.html {{math|''M''<sub>107</sub>}}: Fauquembergue or Powers?].</ref>
| Lucas sequences
|-
| style="text-align:right;"| 12
| style="text-align:right;"| 127
| style="text-align:right;"| 170141183460...715884105727
| style="text-align:right;"| 39
| 1876 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;"| 686479766013...291115057151
| 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>&nbsp;−&nbsp;1 and 2<sup>607</sup>&nbsp;−&nbsp;1 on January 30, 1952." D. H. Lehmer, ''Recent Discoveries of Large Primes'', Mathematics of Computation, vol. 6, No. 37 (1952), p. 61, http://www.ams.org/journals/mcom/1952-06-037/S0025-5718-52-99404-0/S0025-5718-52-99404-0.pdf [Retrieved 2012-09-18]</ref>
| [[Raphael M. Robinson]]
| [[Lucas–Lehmer primality test|LLT]] / [[SWAC (computer)|SWAC]]
|-
| style="text-align:right;"| 14
| style="text-align:right;"| 607
| style="text-align:right;"| 531137992816...219031728127
| 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;"| 104079321946...703168729087
| 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>&nbsp;−&nbsp;1 on June 25. The SWAC tests this number in 13 minutes and 25 seconds." D. H. Lehmer, ''A New Mersenne Prime'', Mathematics of Computation, vol. 6, No. 39 (1952), p. 205, http://www.ams.org/journals/mcom/1952-06-039/S0025-5718-52-99387-3/S0025-5718-52-99387-3.pdf [Retrieved 2012-09-18]</ref>
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 16
| style="text-align:right;"| 2,203
| style="text-align:right;"| 147597991521...686697771007
| style="text-align:right;"| 664
| 1952 October 7<ref name="autogenerated2">"Two more Mersenne primes, 2<sup>2203</sup>&nbsp;−&nbsp;1 and 2<sup>2281</sup>&nbsp;−&nbsp;1, were discovered by the SWAC on October 7 and 9, 1952." D. H. Lehmer, ''Two New Mersenne Primes'', Mathematics of Computation, vol. 7, No. 41 (1952), p. 72, http://www.ams.org/journals/mcom/1953-07-041/S0025-5718-53-99371-5/S0025-5718-53-99371-5.pdf [Retrieved 2012-09-18]</ref>
| Raphael M. Robinson
| LLT / SWAC
|-
| style="text-align:right;"| 17
| style="text-align:right;"| 2,281
| style="text-align:right;"| 446087557183...418132836351
| 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;"| 259117086013...362909315071
| style="text-align:right;"| 969
| 1957 September 8<ref>"On September 8, 1957, the Swedish electronic computer BESK established that the Mersenne number {{nowrap|{{math|''M''<sub>3217</sub>}} {{=}} 2<sup>3217</sup> &minus; 1}} is a prime." Hans Riesel, ''A New Mersenne Prime'', Mathematics of Computation, vol. 12 (1958), p. 60, http://www.ams.org/journals/mcom/1958-12-061/S0025-5718-1958-0099752-6/S0025-5718-1958-0099752-6.pdf [Retrieved 2012-09-18]</ref>
| [[Hans Riesel]]
| LLT / [[BESK]]
|-
| style="text-align:right;"| 19
| style="text-align:right;"| 4,253
| style="text-align:right;"| 190797007524...815350484991
| style="text-align:right;"| 1,281
| 1961 November 3<ref name="Hurwitz and Selfridge 1961">A. Hurwitz and J. L. Selfridge, ''Fermat numbers and perfect numbers'', Notices of the American Mathematical Society, v. 8, 1961, p. 601, abstract 587-104.</ref><ref name="autogenerated5">"If {{math|''p''}} is prime, {{math|''M<sub>p</sub>'' {{=}} 2<sup>''p''</sup> − 1}} is called a Mersenne number. The primes {{math|''M''<sub>4253</sub>}} and {{math|''M''<sub>4423</sub>}} were discovered by coding the Lucas-Lehmer test for the IBM 7090." Alexander Hurwitz, ''New Mersenne Primes'', Mathematics of Computation, vol. 16, No. 78 (1962), pp. 249–251, http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0146162-X/S0025-5718-1962-0146162-X.pdf [Retrieved 2012-09-18]</ref>
| Alexander Hurwitz
| LLT / [[IBM 7090]]
|-
| style="text-align:right;"| 20
| style="text-align:right;"| 4,423
| style="text-align:right;"| 285542542228...902608580607
| style="text-align:right;"| 1,332
| 1961 November 3<ref name="Hurwitz and Selfridge 1961" /><ref name="autogenerated5" />
| Alexander Hurwitz
| LLT / IBM 7090
|-
| style="text-align:right;"| 21
| style="text-align:right;"| 9,689
| style="text-align:right;"| 478220278805...826225754111
| style="text-align:right;"| 2,917
| 1963 May 11<ref name="autogenerated3">"The primes {{math|''M''<sub>9689</sub>}}, {{math|''M''<sub>9941</sub>}}, and {{math|''M''<sub>11213</sub>}} which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois." Donald B. Gillies, ''Three New Mersenne Primes and a Statistical Theory'', Mathematics of Computation, vol. 18, No. 85 (1964), pp. 93–97, http://www.ams.org/journals/mcom/1964-18-085/S0025-5718-1964-0159774-6/S0025-5718-1964-0159774-6.pdf [Retrieved 2012-09-18]</ref>
| [[Donald B. Gillies]]
| LLT / [[ILLIAC II]]
|-
| style="text-align:right;"| 22
| style="text-align:right;"| 9,941
| style="text-align:right;"| 346088282490...883789463551
| 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;"| 281411201369...087696392191
| 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;"| 431542479738...030968041471
| style="text-align:right;"| 6,002
| 1971 March 4<ref>"On the evening of March 4, 1971, a zero Lucas-Lehmer residue for {{math|''p'' {{=}} ''p''<sub>24</sub> {{=}} 19937}} was found. Hence, {{math|''M''<sub>19937</sub>}} is the 24th Mersenne prime." Bryant Tuckerman, ''The 24th Mersenne Prime'', Proceedings of the National Academy of Sciences of the United States of America, vol. 68:10 (1971), pp. 2319–2320, http://www.pnas.org/content/68/10/2319.full.pdf [Retrieved 2012-09-18]</ref>
| [[Bryant Tuckerman]]
| LLT / [[IBM 360]]/91
|-
| style="text-align:right;"| 25
| style="text-align:right;"| 21,701
| style="text-align:right;"| 448679166119...353511882751
| style="text-align:right;"| 6,533
| 1978 October 30<ref>"On October 30, 1978 at 9:40 pm, we found {{math|''M''<sub>21701</sub>}} to be prime. The CPU time required for this test was 7:40:20. Tuckerman and Lehmer later provided confirmation of this result." Curt Noll and Laura Nickel, ''The 25th and 26th Mersenne Primes'', Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]</ref>
| [[Landon Curt Noll]] & Laura Nickel
| LLT / [[CDC Cyber]] 174
|-
| style="text-align:right;"| 26
| style="text-align:right;"| 23,209
| style="text-align:right;"| 402874115778...523779264511
| style="text-align:right;"| 6,987
| 1979 February 9<ref>"Of the 125 remaining {{math|''M<sub>p</sub>''}} only {{math|''M''<sub>23209</sub>}} was found to be prime. The test was completed on February 9, 1979 at 4:06 after 8:39:37 of CPU time. Lehmer and McGrogan later confirmed the result." Curt Noll and Laura Nickel, ''The 25th and 26th Mersenne Primes'', Mathematics of Computation, vol. 35, No. 152 (1980), pp. 1387–1390, http://www.ams.org/journals/mcom/1980-35-152/S0025-5718-1980-0583517-4/S0025-5718-1980-0583517-4.pdf [Retrieved 2012-09-18]</ref>
| Landon Curt Noll
| LLT / CDC Cyber 174
|-
| style="text-align:right;"| 27
| style="text-align:right;"| 44,497
| style="text-align:right;"| 854509824303...961011228671
| 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>&nbsp;–&nbsp;1. [...] Its primeness was determined on April 8, 1979 using the Lucas–Lehmer test. The test was programmed on a CRAY-1 computer by David Slowinski & Harry Nelson." (p. 15) "The result was that after applying the Lucas–Lehmer test to about a thousand numbers, the code determined, on Sunday, April 8th, that 2<sup>44497</sup>&nbsp;−&nbsp;1 is, in fact, the 27th Mersenne prime." (p. 17), David Slowinski, "Searching for the 27th Mersenne Prime",  ''Cray Channels'', vol. 4, no. 1, (1982), pp. 15–17.</ref>
| [[Harry L. Nelson]] & [[David Slowinski]]
| LLT / [[Cray 1]]
|-
| style="text-align:right;"| 28
| style="text-align:right;"| 86,243
| style="text-align:right;"| 536927995502...709433438207
| 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;"| 521928313341...083465515007
| 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 approximately 11 minutes on the SX-2. The discovery of {{math|''M''<sub>110503</sub>}} (January 29, 1988) has been confirmed." W. N. Colquitt and L. Welsh, Jr., ''A New Mersenne Prime'', Mathematics of Computation, vol. 56, No. 194 (April 1991), pp. 867–870, http://www.ams.org/journals/mcom/1991-56-194/S0025-5718-1991-1068823-9/S0025-5718-1991-1068823-9.pdf [Retrieved 2012-09-18]</ref><ref>"This week, two computer experts found the 31st Mersenne prime. But to their surprise, the newly discovered prime number falls between two previously known Mersenne primes. It occurs when {{math|''p'' {{=}} 110,503}}, making it the third-largest Mersenne prime known." I. Peterson, ''Priming for a lucky strike'' Science News; 2/6/88, Vol. 133 Issue 6, pp. 85–85. http://ehis.ebscohost.com/ehost/detail?vid=3&hid=23&sid=9a9d7493-ffed-410b-9b59-b86c63a93bc4%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8824187 [Retrieved 2012-09-18]</ref>
| Walter Colquitt & Luke Welsh
| LLT / [[NEC SX architecture|NEC SX-2]]<ref>{{cite web|url=http://wwwhomes.uni-bielefeld.de/achim/mersenne.html |title=Mersenne Prime Numbers |publisher=Omes.uni-bielefeld.de |date=2011-01-05 |accessdate=2011-05-21}}</ref>
|-
| style="text-align:right;"| 30
| style="text-align:right;"| 132,049
| style="text-align:right;"| 512740276269...455730061311
| style="text-align:right;"| 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. [that is, 1983 September 19]" [https://news.google.com/newspapers?id=roFQAAAAIBAJ&sjid=VRIEAAAAIBAJ&pg=1717%2C4841183 Jim Higgins, "Elusive numeral's number is up" and "Scientist finds big number" in ''The Milwaukee Sentinel'' – Sep 24, 1983, p. 1, ] [https://news.google.com/newspapers?id=roFQAAAAIBAJ&sjid=VRIEAAAAIBAJ&pg=4379%2C4887110 p. 11] [retrieved 2012-10-23]</ref>
| David Slowinski
| LLT / [[Cray X-MP]]
|-
| style="text-align:right;"| 31
| style="text-align:right;"| 216,091
| style="text-align:right;"| 746093103064...103815528447
| 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 {{math|2<sup>''p''</sup> &minus; 1}}, where the exponent {{math|''p''}} is also a prime number. For instance, 127 is a Mersenne number for which the exponent is 7. The record prime number's exponent is 216,091." I. Peterson, ''Prime time for supercomputers'' Science News; 9/28/85, Vol. 128 Issue 13, p. 199. http://ehis.ebscohost.com/ehost/detail?vid=4&hid=22&sid=c11090a2-4670-469f-8f75-947b593a56a0%40sessionmgr10&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=afh&AN=8840537 [Retrieved 2012-09-18]</ref><ref>"Slowinski's program also found the 28th in 1982, the 29th in 1983, and the 30th [known at that time] this past Labor Day weekend. [that is, 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;"| 174135906820...328544677887
| 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;"| 129498125604...243500142591
| style="text-align:right;"| 258,716
| 1994 January 4<ref>Chris Caldwell, [https://web.archive.org/web/19981202090207/http://www.math.unicaen.fr/%7Ereyssat/largest.html The Largest Known Primes].</ref><ref>[http://www.0x07bell.net/WWWMASTER/CrayWWWStuff/prfoldercomp/PRIME_NUMBER.940110.txt Crays press release]</ref><ref>{{cite web|url=https://groups.google.com/forum/#!msg/sci.crypt/XXNkt07IE2s/eDeaWg5-mtIJ|title=Slowinskis email}}</ref>
| David Slowinski & Paul Gage
| LLT / [[Cray C90]]
|-
| style="text-align:right;"| 34
| style="text-align:right;"| 1,257,787
| style="text-align:right;"| 412245773621...976089366527
| style="text-align:right;"| 378,632
| 1996 September 3<ref>Silicon Graphics' press release https://web.archive.org/web/19970606011821/http://www.sgi.com/Headlines/1996/September/prime.html [Retrieved 2012-09-20]</ref>
| David Slowinski & Paul Gage<ref>The Prime Pages, [http://primes.utm.edu/notes/1257787.html A Prime of Record Size! 2<sup>1257787</sup>&nbsp;–&nbsp;1].</ref>
| LLT / [[Cray T90|Cray T94]]
|-
| style="text-align:right;"| 35
| style="text-align:right;"| 1,398,269
| style="text-align:right;"| 814717564412...868451315711
| 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&nbsp;MHz [[Pentium]]
|-
| style="text-align:right;"| 36
| style="text-align:right;"| 2,976,221
| style="text-align:right;"| 623340076248...743729201151
| 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&nbsp;MHz Pentium
|-
| style="text-align:right;"| 37
| style="text-align:right;"| 3,021,377
| style="text-align:right;"| 127411683030...973024694271
| 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&nbsp;MHz Pentium
|-
| style="text-align:right;"| 38
| style="text-align:right;"| 6,972,593
| style="text-align:right;"| 437075744127...142924193791
| 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&nbsp;MHz [[Pentium II]] [[IBM Aptiva]]
|-
| style="text-align:right;"| 39
| style="text-align:right;"| 13,466,917
| style="text-align:right;"| 924947738006...470256259071
| 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&nbsp;MHz [[Athlon Thunderbird|Athlon T-Bird]]
|-
| style="text-align:right;"| 40
| style="text-align:right;"| 20,996,011
| style="text-align:right;"| 125976895450...762855682047
| 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&nbsp;GHz [[Dell Dimension]]
|-
| style="text-align:right;"| 41
| style="text-align:right;"| 24,036,583
| style="text-align:right;"| 299410429404...882733969407
| 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>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2.4&nbsp;GHz [[Pentium 4]]
|-
| style="text-align:right;"| 42
| style="text-align:right;"| 25,964,951
| style="text-align:right;"| 122164630061...280577077247
| 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>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2.4&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 43
| style="text-align:right;"| 30,402,457
| style="text-align:right;"| 315416475618...411652943871
| 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>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 2&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 44
| style="text-align:right;"| 32,582,657
| style="text-align:right;"| 124575026015...154053967871
| 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>&nbsp;–&nbsp;1].</ref>
| LLT / Prime95 on 3&nbsp;GHz Pentium 4
|-
| style="text-align:right;"| 45
| style="text-align:right;"| 37,156,667
| style="text-align:right;"| 202254406890...022308220927
| 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&nbsp;GHz [[Core 2 Duo]]
|-
| style="text-align:right;"| 46
| style="text-align:right;"| 42,643,801
| style="text-align:right;"| 169873516452...765562314751
| style="text-align:right;"| 12,837,064
| 2009 June 4<ref group="n" name="date_issue">Although {{math|''M''<sub>42,643,801</sub>}} was first reported by a machine on April 12, 2009, no human took notice of this fact until June 4, 2009.</ref>
| GIMPS / Odd M. Strindmo<ref>"On April 12th [2009], the 47th known Mersenne prime, 2<sup>42,643,801</sup>&nbsp;–&nbsp;1, a 12,837,064 digit number was found by Odd Magnar Strindmo from Melhus, Norway! This prime is the second largest known prime number, a "mere" 141,125 digits smaller than the Mersenne prime found last August.", ''The List of Largest Known Primes Home Page'', http://primes.utm.edu/primes/page.php?id=88847 [retrieved 2012-09-18]</ref><ref group="n" name="Strindmo">Strindmo also uses the alias Stig M. Valstad.</ref>
| LLT / Prime95 on 3&nbsp;GHz Core 2
|-
| style="text-align:right;"| 47
| style="text-align:right;"| 43,112,609
| style="text-align:right;"| 316470269330...166697152511
| 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 group="n" name="unverified_index">It is not verified whether any undiscovered Mersenne primes exist between the 47th ({{math|''M''<sub>43,112,609</sub>}}) and the 51st ({{math|''M''<sub>82,589,933</sub>}}) on this chart; the ranking is therefore provisional.</ref>
| style="text-align:right;"| 57,885,161
| style="text-align:right;"| 581887266232...071724285951
| style="text-align:right;"| 17,425,170
| 2013 January 25
| GIMPS / Curtis Cooper<ref name="m48">{{cite web|url=http://mersenne.org/various/57885161.htm|title=GIMPS Discovers 48th Mersenne Prime, 2<sup>57,885,161</sup> &minus; 1 is now the Largest Known Prime.|publisher=[[Great Internet Mersenne Prime Search]]|accessdate=2016-01-19}}</ref>
| LLT / Prime95 on 3&nbsp;GHz Intel Core2 Duo E8400<ref>{{cite web|url=http://www.mersenne.org/primes/|title=List of known Mersenne prime numbers|accessdate=29 November 2014}}</ref>
|-
| style="text-align:right;"| 49<ref group="n" name="unverified_index" />
| style="text-align:right;"| 74,207,281
| style="text-align:right;"| 300376418084...391086436351
| style="text-align:right;"| 22,338,618
| 2016 January 7<ref group="n" name="date_issue_2>Although {{math|''M''<sub>74,207,281</sub>}} was first reported by a machine on September 17, 2015, no human took notice of this fact until January 7, 2016.</ref>
| GIMPS / Curtis Cooper<ref name="GIMPS-2016" />
| LLT / Prime95 on Intel [[Core i7-4790]]
|-
| style="text-align:right;"| 50<ref group="n" name="unverified_index" />
| style="text-align:right;"| 77,232,917
| style="text-align:right;"| 467333183359...069762179071
| style="text-align:right;"| 23,249,425
| 2017 December 26
| GIMPS / Jon Pace<ref name="GIMPS-2017">{{cite web |title=GIMPS Project Discovers Largest Known Prime Number: 2<sup>77,232,917</sup>-1 |url=https://www.mersenne.org/primes/press/M77232917.html |date=3 January 2018 |work=Mersenne Research, Inc. |accessdate=3 January 2018 }}</ref>
| LLT / Prime95 on 3.3&nbsp;GHz Intel [[Core i5-6600]]<ref>{{cite web|url=https://www.mersenne.org/primes/|title=List of known Mersenne prime numbers|accessdate=3 January 2018}}</ref>
|-
| style="text-align:right;"| 51<ref group="n" name="unverified_index" />
| style="text-align:right;"| 82,589,933
| style="text-align:right;"| 148894445742...325217902591
| style="text-align:right;"| 24,862,048
| 2018 December 7
| GIMPS / Patrick Laroche<ref name="GIMPS-2018" />
| LLT / Prime95 on Intel [[Core i5-4590T]]
|}

<references group="n" />

All Mersenne numbers below the 51st Mersenne prime ({{math|''M''<sub>82,589,933</sub>}}) have been tested at least once but some have not been double-checked. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered ''after'' the 30th and the 31st. Similarly, {{math|''M''<sub>43,112,609</sub>}} was followed by two smaller Mersenne primes, first 2 weeks later and then 9 months later.<ref name="GIMPS Milestones">[https://www.mersenne.org/report_milestones/ GIMPS Milestones Report]. Retrieved 2019-05-17</ref> {{math|''M''<sub>43,112,609</sub>}} was the first discovered prime number with more than 10 million decimal digits.

The largest known Mersenne prime {{nowrap|(2<sup>82,589,933</sup> − 1)}} is also the [[largest known prime number]].<ref name="GIMPS-2018" />

The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992.<ref>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==
Since they are prime numbers, Mersenne primes are divisible only by 1 and by themselves.  However, not all Mersenne numbers are Mersenne primes, and the composite Mersenne numbers may be factored non-trivially.  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|2019|6}}, 2{{sup|1,193}}&nbsp;−&nbsp;1 is the record-holder,<ref>[[Thorsten Kleinjung]], [[Joppe Bos]], [[Arjen Lenstra]] "Mersenne Factorization Factory" http://eprint.iacr.org/2014/653.pdf</ref> having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once.  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|2019|6}}, the largest factorization with [[probable prime]] factors allowed is {{nowrap|2<sup>7,313,983</sup> &minus; 1 {{=}} 305,492,080,276,193 × {{math|''q''}}}}, where {{math|''q''}} is a 2,201,714-digit probable prime. It was discovered by Oliver Kruse.<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=2018-03-21}}</ref>  {{As of|2019|6}}, the Mersenne number ''M''<sub>1277</sub> is the smallest composite Mersenne number with no known factors; it has no prime factors below 2<sup>67</sup>.<ref>{{cite web|url=https://www.mersenne.org/report_exponent/?exp_lo=1277&full=1| title = Exponent Status for M1277|accessdate=2018-06-22}}</ref>

The table below shows factorizations for the first 20 composite Mersenne numbers {{OEIS|id=A244453}}.

{|class="wikitable"
!{{math|''p''}}
!{{math|''M<sub>p</sub>''}}
!Factorization of {{math|''M<sub>p</sub>''}}
|-
|11
|2047
|23 × 89
|-
|23
|8388607
|47 × 178,481
|-
|29
|536870911
|233 × 1,103 × 2,089
|-
|37
|137438953471
|223 × 616,318,177
|-
|41
|2199023255551
|13,367 × 164,511,353
|-
|43
|8796093022207
|431 × 9,719 × 2,099,863
|-
|47
|140737488355327
|2,351 × 4,513 × 13,264,529
|-
|53
|9007199254740991
|6,361 × 69,431 × 20,394,401
|-
|59
|57646075230343487
|179,951 × 3,203,431,780,337 (13 digits)
|-
|67
|147573952589676412927
|193,707,721 × 761,838,257,287 (12 digits)
|-
|71
|2361183241434822606847
|228,479 × 48,544,121 × 212,885,833
|-
|73
|9444732965739290427391
|439 × 2,298,041 × 9,361,973,132,609 (13 digits)
|-
|79
|604462903807314587353087
|2,687 × 202,029,703 × 1,113,491,139,767 (13 digits)
|-
|83
|967140655691...033397649407
|167 × 57,912,614,113,275,649,087,721 (23 digits)
|-
|97
|158456325028...187087900671
|11,447 × 13,842,607,235,828,485,645,766,393 (26 digits)
|-
|101
|253530120045...993406410751
|7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits)
|-
|103
|101412048018...973625643007
|2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits)
|-
|109
|649037107316...312041152511
|745,988,807 × 870,035,986,098,720,987,332,873 (24 digits)
|-
|113
|103845937170...992658440191
|3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits)
|-
|131
|272225893536...454145691647
|263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits)
|}

The number of factors for the first 500 Mersenne numbers can be found at {{OEIS|id=A046800}}.

==Mersenne numbers in nature and elsewhere==
In the mathematical problem [[Tower of Hanoi]], solving a puzzle with an {{math|''n''}}-disc tower requires {{math|''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=978-0-8218-4814-2|pages=197}}</ref> The number of rice grains on the whole chessboard in the [[wheat and chessboard problem]] is {{math|''M<sub>64</sub>''}}.

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>

In [[geometry]], an integer [[right triangle]] that is [[Pythagorean triple|primitive]] and has its even leg a power of 2 (&nbsp;{{math|&ge; 4}}&nbsp;) generates a unique right triangle such that its [[inradius]] is always a Mersenne number. For example, if the even leg is {{math|2<sup>''n''&nbsp;+&nbsp;1</sup>}} then because it is primitive it constrains the odd leg to be {{math|4<sup>''n''</sup>&nbsp;&minus;&nbsp;1}}, the [[hypotenuse]] to be {{math|4<sup>''n''</sup>&nbsp;+&nbsp;1}} and its inradius to be {{math|2<sup>''n''</sup>&nbsp;&minus;&nbsp;1}}.<ref>{{cite web |url=http://oeis.org/A016131 |title=OEIS A016131 |publisher=The On-Line Encyclopedia of Integer Sequences}}</ref>

The Mersenne numbers were studied with respect to the total number of accepting paths of [[Non-deterministic Turing machine|non-deterministic polynomial time Turing machines]] in 2018<ref>{{cite web |url=https://eccc.weizmann.ac.il/report/2018/166 |author= Tayfun Pay, and James L. Cox |title=An overview of some semantic and syntactic complexity classes}}</ref> and intriguing inclusions were discovered.

==Mersenne–Fermat primes==
A '''Mersenne–Fermat number''' is defined as {{math|{{sfrac|2''<sup>p<sup>r</sup></sup>'' &minus; 1|2<sup>''p''<sup>''r'' &minus; 1</sup></sup> &minus; 1}}}}, with {{math|''p''}} prime, {{math|''r''}} natural number, and can be written as '''{{math|MF(''p'', ''r'')}}''', when {{math|''r'' {{=}} 1}}, it is a Mersenne number, and when {{math|''p'' {{=}} 2}}, it is a [[Fermat number]], the only known Mersenne–Fermat prime with {{math|''r'' > 1}} are
:{{math|MF(2, 2), MF(3, 2), MF(7, 2), MF(59, 2), MF(2, 3), MF(3, 3), MF(2, 4),}} and {{math|MF(2, 5)}}.<ref>{{cite web|url=http://staff.spd.dcu.ie/johnbcos/fermat6.htm|title=A research of Mersenne and Fermat primes|url-status=dead|archiveurl=https://archive.is/20120529014233/http://staff.spd.dcu.ie/johnbcos/fermat6.htm|archivedate=2012-05-29}}</ref>

In fact, {{math|MF(''p'', ''r'') {{=}} ''Φ<sub>p<sup>r</sup></sub>''(2)}}, where {{math|''Φ''}} is the [[cyclotomic polynomial]].

==Generalizations==
The simplest generalized Mersenne primes are prime numbers of the form {{math|''f''(2<sup>''n''</sup>)}}, where {{math|''f''(''x'')}} is a low-degree [[polynomial]] with small integer [[coefficients]].<ref>{{cite book|title=Encyclopedia of Cryptography and Security|first=Jerome A.|last=Solinas|editor-first1=Henk C. A. van|editor-last1=Tilborg|editor-first2=Sushil|editor-last2=Jajodia|date=1 January 2011|publisher=Springer US|pages=509–510|doi=10.1007/978-1-4419-5906-5_32|chapter = Generalized Mersenne Prime|isbn = 978-1-4419-5905-8}}</ref> An example is {{math|2<sup>64</sup> − 2<sup>32</sup> + 1}}, in this case, {{math|''n'' &#61; 32}}, and {{math|''f''(''x'') &#61; ''x''<sup>2</sup> − ''x'' + 1}}; another example is {{math|2<sup>192</sup> − 2<sup>64</sup> − 1}}, in this case, {{math|''n'' &#61; 64}}, and {{math|''f''(''x'') &#61; ''x''<sup>3</sup> − ''x'' − 1}}.

It is also natural to try to generalize primes of the form {{math|2<sup>''n''</sup> − 1}} to primes of the form {{math|''b''<sup>''n''</sup> − 1}} (for {{math|''b'' ≠ 2}} and {{math|''n'' > 1}}). However (see also [[#Theorems about Mersenne numbers|theorems above]]), {{math|''b''<sup>''n''</sup> − 1}} is always divisible by {{math|''b'' − 1}}, so unless the latter is a [[unit (ring theory)|unit]], the former is not a prime. This can be remedied by allowing ''b'' to be an algebraic integer instead of an integer:

===Complex numbers===
In the [[ring (mathematics)|ring]] of integers (on [[real number]]s), if {{math|''b'' − 1}} is a [[unit (ring theory)|unit]], then {{math|''b''}} is either 2 or 0. But {{math|2<sup>''n''</sup> − 1}} are the usual Mersenne primes, and the formula {{math|0<sup>''n''</sup> − 1}} does not lead to anything interesting (since it is always −1 for all {{math|''n'' > 0}}). Thus, we can regard a ring of "integers" on [[complex number]]s instead of [[real number]]s, like [[Gaussian integer]]s and [[Eisenstein integer]]s.

====Gaussian Mersenne primes====
If we regard the ring of [[Gaussian integer]]s, we get the case {{math|''b'' {{=}} 1 + ''i''}} and {{math|''b'' {{=}} 1 − ''i''}}, and can ask ([[without loss of generality|WLOG]]) for which {{math|''n''}} the number {{math|(1 + ''i'')<sup>''n''</sup> − 1}} is a [[Gaussian prime]] which will then be called a '''Gaussian Mersenne prime'''.<ref>Chris Caldwell: [http://primes.utm.edu/glossary/xpage/GaussianMersenne.html The Prime Glossary: Gaussian Mersenne] (part of the [[Prime Pages]])</ref>

{{math|(1 + ''i'')<sup>''n''</sup> − 1}} is a Gaussian prime for the following {{math|''n''}}:

:2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... {{OEIS|id=A057429}}

Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.

As for all Gaussian primes, the [[Gaussian integer#Norm of a Gaussian integer|norm]]s (that is, squares of absolute values) of these numbers are rational primes:

:5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... {{OEIS|id=A182300}}.

====Eisenstein Mersenne primes====
We can also regard the ring of [[Eisenstein integer]]s,  we get the case {{math|''b'' {{=}} 1 + ''ω''}} and {{math|''b'' {{=}} 1 − ''ω''}}, and can ask for what {{math|''n''}} the number {{math|(1 + ''ω'')<sup>''n''</sup> − 1}} is an ''Eisenstein prime'' which will then be called a '''Eisenstein Mersenne prime'''.

{{math|(1 + ''ω'')<sup>''n''</sup> − 1}} is an Eisenstein prime for the following {{math|''n''}}:

:2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... {{OEIS|id=A066408}}

The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:

:7, 271, 2269, 176419, 129159847, 1162320517, ... {{OEIS|id=A066413}}

===Divide an integer===
====Repunit primes====
{{main|Repunit}}
The other way to deal with the fact that {{math|''b''<sup>''n''</sup> − 1}} is always divisible by {{math|''b'' − 1}}, it is to simply take out this factor and ask which values of {{math|''n''}} make
:<math>\frac{b^n-1}{b-1}</math>
be prime. (The integer {{math|''b''}} can be either positive or negative.) If, for example, we take {{math|''b'' {{=}} 10}}, we get {{math|''n''}} values of:
:2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... {{OEIS|id=A004023}},<br>corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... {{OEIS|id=A004022}}.
These primes are called '''[[repunit]] primes'''. Another example is when we take {{math|''b'' {{=}} −12}}, we get {{math|''n''}} values of:
:2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... {{OEIS|id=A057178}},<br>corresponding to primes −11, 19141, 57154490053, ....
It is a conjecture that for every integer {{math|''b''}} which is not a [[perfect power]], there are infinitely many values of {{math|''n''}} such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime. (When {{math|''b''}} is a perfect power, it can be shown that there is at most one {{math|''n''}} value such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime)

Least {{math|''n''}} such that {{math|{{sfrac|''b''<sup>''n''</sup> − 1|''b'' − 1}}}} is prime are (starting with {{math|''b'' {{=}} 2}}, {{math|0}} if no such {{math|''n''}} exists)
:2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... {{OEIS|id=A084740}}

For negative bases {{math|''b''}}, they are (starting with {{math|''b'' {{=}} −2}}, {{math|0}} if no such {{math|''n''}} exists)
:3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... {{OEIS|id=A084742}} (notice this OEIS sequence does not allow {{math|''n'' {{=}} 2}})

Least base {{math|''b''}} such that {{math|{{sfrac|''b''<sup>prime(''n'')</sup> − 1|''b'' − 1}}}} is prime are
:2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... {{OEIS|id=A066180}}

For negative bases {{math|''b''}}, they are
:3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... {{OEIS|id=A103795}}

====Other generalized Mersenne primes====
Another generalized Mersenne number is
:<math>\frac{a^n-b^n}{a-b}</math>
with {{math|''a''}}, {{math|''b''}} any [[coprime]] integers, {{math|''a'' > 1}} and {{math|−''a'' < ''b'' < ''a''}}. (Since {{math|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>}} is always divisible by {{math|''a'' − ''b''}}, the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the [[Lucas sequence|Lucas number]] {{math|''U<sub>n</sub>''(''a'' + ''b'', ''ab'')}}, since {{math|''a''}} and {{math|''b''}} are the [[zero of a function|roots]] of the [[quadratic equation]] {{math|''x''<sup>2</sup> − (''a'' + ''b'')''x'' + ''ab'' {{=}} 0}}, and this number equals 1 when {{math|''n'' {{=}} 1}}) We can ask which {{math|''n''}} makes this number prime.  It can be shown that such {{math|''n''}} must be primes themselves or equal to 4, and {{math|''n''}} can be 4 if and only if {{math|''a'' + ''b'' {{=}} 1}} and {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}} is prime. (Since {{math|{{sfrac|''a''<sup>4</sup> − ''b''<sup>4</sup>|''a'' − ''b''}} {{=}} (''a'' + ''b'')(''a''<sup>2</sup> + ''b''<sup>2</sup>)}}. Thus, in this case the pair {{math|(''a'', ''b'')}} must be {{math|(''x'' + 1, −''x'')}} and {{math|''x''<sup>2</sup> + (''x'' + 1)<sup>2</sup>}} must be prime. That is, {{math|''x''}} must be in {{oeis|id=A027861}}.) It is a conjecture that for any pair {{math|(''a'', ''b'')}} such that for every natural number {{math|''r'' > 1}}, {{math|''a''}} and {{math|''b''}} are not both perfect {{math|''r''}}th powers, and {{math|−4''ab''}} is not a perfect [[fourth power]]. there are infinitely many values of {{math|''n''}} such that {{math|{{sfrac|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>|''a'' − ''b''}}}} is prime.  (When {{math|''a''}} and {{math|''b''}} are both perfect {{math|''r''}}th powers for an {{math|''r'' > 1}} or when {{math|−4''ab''}} is a perfect fourth power, it can be shown that there are at most two {{math|''n''}} values with this property, since if so, then {{math|{{sfrac|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>|''a'' − ''b''}}}} can be factored algebraically) However, this has not been proved for any single value of {{math|(''a'', ''b'')}}.

{|class="wikitable
|+For more information, see <ref>{{cite paper |first=Ali |last=Zalnezhad |first2=Hossein |last2=Zalnezhad |first3=Ghasem |last3=Shabani |first4=Mehdi |last4=Zalnezhad |title=Relationships and Algorithm in order to Achieve the Largest Primes |date=March 2015 |arxiv=1503.07688 }}</ref><ref>[http://www.primenumbers.net/Henri/us/MersFermus.htm {{math|(''x'', 1)}} and {{math|(''x'', −1)}} for {{math|''x''}} = 2 to 50]</ref><ref>[http://www.fermatquotient.com/PrimSerien/GenRepu.txt {{math|(''x'', 1)}} for {{math|''x''}} = 2 to 160]</ref><ref>[http://www.fermatquotient.com/PrimSerien/GenRepuP.txt {{math|(''x'', −1)}} for {{math|''x''}} = 2 to 160]</ref><ref>[http://www.fermatquotient.com/PrimSerien/PrimPot.txt {{math|(''x'' + 1, ''x'')}} for {{math|''x''}} = 1 to 160]</ref><ref>[http://www.fermatquotient.com/PrimSerien/PrimPotP.txt {{math|(''x'' + 1, −''x'')}} for {{math|''x''}} = 1 to 40]</ref><ref>[http://www.fermatquotient.com/PrimSerien/PrimPot2.txt {{math|(''x'' + 2, ''x'')}} for odd {{math|''x''}} = 1 to 107]</ref><ref>[https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf#search=%22107%20811%202819%204817%209601%22 {{math|(''x'', −1)}} for {{math|''x''}} = 2 to 200]</ref><ref>[http://www.primenumbers.net/prptop/searchform.php?form=%28a%5En-b%5En%29%2Fc&action=Search PRP records, search for (a^n-b^n)/c, that is, {{math|(''a'', ''b'')}}]</ref><ref>[http://www.primenumbers.net/prptop/searchform.php?form=%28a%5En%2Bb%5En%29%2Fc&action=Search PRP records, search for (a^n+b^n)/c, that is, {{math|(''a'', −''b'')}}]</ref>
!{{math|''a''}}
!{{math|''b''}}
!numbers {{math|''n''}} such that {{math|{{sfrac|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>|''a'' − ''b''}}}} is prime<br/>(some large terms are only [[probable prime]]s, these {{math|''n''}} are checked up to 100000 for {{math|{{abs|''b''}} ≤ 5}} or {{math|{{abs|''b''}} {{=}} ''a'' − 1}}, 20000 for {{math|5 < {{abs|''b''}} < ''a'' − 1}})
![[OEIS]] sequence
|-
|2
|1
|2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, ..., 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ...
|{{OEIS link|id=A000043}}
|-
|2
|−1
|3, 4<sup>*</sup>, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... 
|{{OEIS link|id=A000978}}
|-
|3
|2
|2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ...
|{{OEIS link|id=A057468}}
|-
|3
|1
|3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ...
|{{OEIS link|id=A028491}}
|-
|3
|−1
|2<sup>*</sup>, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ...
|{{OEIS link|id=A007658}}
|-
|3
|−2
|3, 4<sup>*</sup>, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ...
|{{OEIS link|id=A057469}}
|-
|4
|3
|2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ...
|{{OEIS link|id=A059801}}
|-
|4
|1
|2 (no others)
|
|-
|4
|−1
|2<sup>*</sup>, 3 (no others)
|
|-
|4
|−3
|3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ...
|{{OEIS link|id=A128066}}
|-
|5
|4
|3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ...
|{{OEIS link|id=A059802}}
|-
|5
|3
|13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ...
|{{OEIS link|id=A121877}}
|-
|5
|2
|2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ...
|{{OEIS link|id=A082182}}
|-
|5
|1
|3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ...
|{{OEIS link|id=A004061}}
|-
|5
|−1
|5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ...
|{{OEIS link|id=A057171}}
|-
|5
|−2
|2<sup>*</sup>, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ...
|{{OEIS link|id=A082387}}
|-
|5
|−3
|2<sup>*</sup>, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ...
|{{OEIS link|id=A122853}}
|-
|5
|−4
|4<sup>*</sup>, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ...
|{{OEIS link|id=A128335}}
|-
|6
|5
|2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ...
|{{OEIS link|id=A062572}}
|-
|6
|1
|2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ...
|{{OEIS link|id=A004062}}
|-
|6
|−1
|2<sup>*</sup>, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ...
|{{OEIS link|id=A057172}}
|-
|6
|−5
|3, 4<sup>*</sup>, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ...
|{{OEIS link|id=A128336}}
|-
|7
|6
|2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ...
|{{OEIS link|id=A062573}}
|-
|7
|5
|3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ...
|{{OEIS link|id=A128344}}
|-
|7
|4
|2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ...
|{{OEIS link|id=A213073}}
|-
|7
|3
|3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ...
|{{OEIS link|id=A128024}}
|-
|7
|2
|3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ...
|{{OEIS link|id=A215487}}
|-
|7
|1
|5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...
|{{OEIS link|id=A004063}}
|-
|7
|−1
|3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ...
|{{OEIS link|id=A057173}}
|-
|7
|−2
|2<sup>*</sup>, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ...
|{{OEIS link|id=A125955}}
|-
|7
|−3
|3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ...
|{{OEIS link|id=A128067}}
|-
|7
|−4
|2<sup>*</sup>, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ...
|{{OEIS link|id=A218373}}
|-
|7
|−5
|2<sup>*</sup>, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ...
|{{OEIS link|id=A128337}}
|-
|7
|−6
|3, 53, 83, 487, 743, ...
|{{OEIS link|id=A187805}}
|-
|8
|7
|7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ...
|{{OEIS link|id=A062574}}
|-
|8
|5
|2, 19, 1021, 5077, 34031, 46099, 65707, ...
|{{OEIS link|id=A128345}}
|-
|8
|3
|2, 3, 7, 19, 31, 67, 89, 9227, 43891, ...
|{{OEIS link|id=A128025}}
|-
|8
|1
|3 (no others)
|
|-
|8
|−1
|2<sup>*</sup> (no others)
|
|-
|8
|−3
|2<sup>*</sup>, 5, 163, 191, 229, 271, 733, 21059, 25237, ...
|{{OEIS link|id=A128068}}
|-
|8
|−5
|2<sup>*</sup>, 7, 19, 167, 173, 223, 281, 21647, ...
|{{OEIS link|id=A128338}}
|-
|8
|−7
|4<sup>*</sup>, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ...
|{{OEIS link|id=A181141}}
|-
|9
|8
|2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ...
|{{OEIS link|id=A059803}}
|-
|9
|7
|3, 5, 7, 4703, 30113, ...
|{{OEIS link|id=A273010}}
|-
|9
|5
|3, 11, 17, 173, 839, 971, 40867, 45821, ...
|{{OEIS link|id=A128346}}
|-
|9
|4
|2 (no others)
|
|-
|9
|2
|2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ...
|{{OEIS link|id=A173718}}
|-
|9
|1
|(none)
|
|-
|9
|−1
|3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ...
|{{OEIS link|id=A057175}}
|-
|9
|−2
|2<sup>*</sup>, 3, 7, 127, 283, 883, 1523, 4001, ...
|{{OEIS link|id=A125956}}
|-
|9
|−4
|2<sup>*</sup>, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ...
|{{OEIS link|id=A211409}}
|-
|9
|−5
|3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ...
|{{OEIS link|id=A128339}}
|-
|9
|−7
|2<sup>*</sup>, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ...
|{{OEIS link|id=A301369}}
|-
|9
|−8
|3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ...
|{{OEIS link|id=A187819}}
|-
|10
|9
|2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ...
|{{OEIS link|id=A062576}}
|-
|10
|7
|2, 31, 103, 617, 10253, 10691, ...
|{{OEIS link|id=A273403}}
|-
|10
|3
|2, 3, 5, 37, 599, 38393, 51431, ...
|{{OEIS link|id=A128026}}
|-
|10
|1
|2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...
|{{OEIS link|id=A004023}}
|-
|10
|−1
|5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ...
|{{OEIS link|id=A001562}}
|-
|10
|−3
|2<sup>*</sup>, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ...
|{{OEIS link|id=A128069}}
|-
|10
|−7
|2<sup>*</sup>, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
|
|-
|10
|−9
|4<sup>*</sup>, 7, 67, 73, 1091, 1483, 10937, ...
|{{OEIS link|id=A217095}}
|-
|11
|10
|3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ...
|{{OEIS link|id=A062577}}
|-
|11
|9
|5, 31, 271, 929, 2789, 4153, ...
|{{OEIS link|id=A273601}}
|-
|11
|8
|2, 7, 11, 17, 37, 521, 877, 2423, ...
|{{OEIS link|id=A273600}}
|-
|11
|7
|5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ...
|{{OEIS link|id=A273599}}
|-
|11
|6
|2, 3, 11, 163, 191, 269, 1381, 1493, ...
|{{OEIS link|id=A273598}}
|-
|11
|5
|5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ...
|{{OEIS link|id=A128347}}
|-
|11
|4
|3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ...
|{{OEIS link|id=A216181}}
|-
|11
|3
|3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ...
|{{OEIS link|id=A128027}}
|-
|11
|2
|2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ...
|{{OEIS link|id=A210506}}
|-
|11
|1
|17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...
|{{OEIS link|id=A005808}}
|-
|11
|−1
|5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ...
|{{OEIS link|id=A057177}}
|-
|11
|−2
|3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ...
|{{OEIS link|id=A125957}}
|-
|11
|−3
|3, 103, 271, 523, 23087, 69833, ...
|{{OEIS link|id=A128070}}
|-
|11
|−4
|2<sup>*</sup>, 7, 53, 67, 71, 443, 26497, ...
|{{OEIS link|id=A224501}}
|-
|11
|−5
|7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ...
|{{OEIS link|id=A128340}}
|-
|11
|−6
|2<sup>*</sup>, 5, 7, 107, 383, 17359, 21929, 26393, ...
|
|-
|11
|−7
|7, 1163, 4007, 10159, ...
|
|-
|11
|−8
|2<sup>*</sup>, 3, 13, 31, 59, 131, 223, 227, 1523, ...
|
|-
|11
|−9
|2<sup>*</sup>, 3, 17, 41, 43, 59, 83, ...
|
|-
|11
|−10
|53, 421, 647, 1601, 35527, ...
|{{OEIS link|id=A185239}}
|-
|12
|11
|2, 3, 7, 89, 101, 293, 4463, 70067, ...
|{{OEIS link|id=A062578}}
|-
|12
|7
|2, 3, 7, 13, 47, 89, 139, 523, 1051, ...
|{{OEIS link|id=A273814}}
|-
|12
|5
|2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ...
|{{OEIS link|id=A128348}}
|-
|12
|1
|2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...
|{{OEIS link|id=A004064}}
|-
|12
|−1
|2<sup>*</sup>, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ...
|{{OEIS link|id=A057178}}
|-
|12
|−5
|2<sup>*</sup>, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ...
|{{OEIS link|id=A128341}}
|-
|12
|−7
|2<sup>*</sup>, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
|
|-
|12
|−11
|47, 401, 509, 8609, ...
|{{OEIS link|id=A213216}}
|}

<sup>*</sup>Note: if {{math|''b'' < 0}} and {{math|''n''}} is even, then the numbers {{math|''n''}} are not included in the corresponding OEIS sequence.

A conjecture related to the generalized Mersenne primes:<ref name="Wagstaff">{{cite web|url=http://primes.utm.edu/mersenne/heuristic.html|title=Heuristics: Deriving the Wagstaff Mersenne Conjecture|first=Chris|last=Caldwell|publisher=}}</ref><ref>{{cite web|url=https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0906&L=NMBRTHRY&P=R295&1=NMBRTHRY&9=A&J=on&d=No%20Match;Match;Matches&z=4|title=Generalized Repunit Conjecture}}</ref> (the conjecture predicts where is the next generalized Mersenne prime, if the conjecture is true, then there are infinitely many primes for all such {{math|(''a'',''b'')}} pairs)

For any integers {{math|''a''}} and {{math|''b''}} which satisfy the conditions:

# {{math|''a'' > 1}}, {{math|−''a'' < ''b'' < ''a''}}.
# {{math|''a''}} and {{math|''b''}} are [[coprime]]. (thus, {{math|''b''}} cannot be 0)
# For every natural number {{math|''r'' > 1}}, {{math|''a''}} and {{math|''b''}} are not both perfect {{math|''r''}}th powers. (since when {{math|''a''}} and {{math|''b''}} are both perfect {{math|''r''}}th powers, it can be shown that there are at most two {{math|''n''}} value such that {{math|{{sfrac|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>|''a'' − ''b''}}}} is prime, and these {{math|''n''}} values are {{math|''r''}} itself or a [[nth root|root]] of {{math|''r''}}, or 2)
# {{math|−4''ab''}} is not a perfect fourth power (if so, then the number has [[aurifeuillean factorization]]).

has prime numbers of the form

:<math>R_p(a,b)=\frac{a^p-b^p}{a-b}</math>

for prime {{math|''p''}}, the prime numbers will be distributed near the best fit line

:<math>Y=G \cdot \log_a(\log_a(R_{(a,b)}(n)))+C</math>

where

: <math> \lim_{n\rightarrow\infty} G=\frac{1}{e^\gamma}=0.561459483566\ldots </math>

and there are about

:<math>(\log_e(N)+m \cdot \log_e(2) \cdot \log_e \left(\log_e(N))+\frac{1}{\sqrt N}-\delta\right) \cdot \frac{e^\gamma}{\log_e(a)}</math>

prime numbers of this form less than {{math|''N''}}.

*{{math|''e''}} is the [[e (mathematical constant)|base of the natural logarithm]].
*{{math|''γ''}} is the [[Euler–Mascheroni constant]].
*{{math|log<sub>''a''</sub>}} is the [[logarithm]] in [[base (exponentiation)|base]] {{math|''a''}}.
*{{math|''R''<sub>(''a'',''b'')</sub>(''n'')}} is the {{math|''n''}}th prime number of the form {{math|{{sfrac|''a''<sup>''p''</sup> − ''b''<sup>''p''</sup>|''a'' − ''b''}}}} for prime {{math|''p''}}.
*{{math|''C''}} is a data fit constant which varies with {{math|''a''}} and {{math|''b''}}.
*{{math|''δ''}} is a data fit constant which varies with {{math|''a''}} and {{math|''b''}}.
*{{math|''m''}} is the largest natural number such that {{math|''a''}} and {{math|−''b''}} are both perfect {{math|2{{sup|''m'' − 1}}}}th powers.

We also have the following three properties:

# The number of prime numbers of the form {{math|{{sfrac|''a''<sup>''p''</sup> − ''b''<sup>''p''</sup>|''a'' − ''b''}}}} (with prime {{math|''p''}}) less than or equal to {{math|''n''}} is about {{math|''e''<sup>''γ''</sup> log<sub>''a''</sub>(log<sub>''a''</sub>(''n''))}}.
# The expected number of prime numbers of the form {{math|{{sfrac|''a''<sup>''p''</sup> − ''b''<sup>''p''</sup>|''a'' − ''b''}}}} with prime {{math|''p''}} between {{math|''n''}} and {{math|''an''}} is about {{math|''e''<sup>''γ''</sup>}}. 
# The probability that number of the form {{math|{{sfrac|''a''<sup>''p''</sup> − ''b''<sup>''p''</sup>|''a'' − ''b''}}}} is prime (for prime {{math|''p''}}) is about {{math|{{sfrac|''e''<sup>''γ''</sup>|''p'' log<sub>''e''</sub>(''a'')}}}}.

If this conjecture is true, then for all such {{math|(''a'',''b'')}} pairs, let {{math|''q''}} be the {{math|''n''}}th prime of the form {{math|{{sfrac|''a''<sup>''p''</sup> − ''b''<sup>''p''</sup>|''a'' − ''b''}}}}, the graph of {{math|log<sub>''a''</sub>(log<sub>''a''</sub>(''q''))}} versus {{math|''n''}} is almost linear. (See <ref name="Wagstaff"/>)

When {{math|''a'' {{=}} ''b'' + 1}}, it is {{math|(''b'' + 1){{sup|''n''}} − ''b''{{sup|''n''}}}}, a difference of two consecutive perfect {{math|''n''}}th powers, and if {{math|''a''<sup>''n''</sup> − ''b''<sup>''n''</sup>}} is prime, then {{math|''a''}} must be {{math|''b'' + 1}}, because it is divisible by {{math|''a'' − ''b''}}.

Least {{math|''n''}} such that {{math|(''b'' + 1){{sup|''n''}} − ''b''{{sup|''n''}}}} is prime are
:2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... {{OEIS|id=A058013}}

Least {{math|''b''}} such that {{math|(''b'' + 1){{sup|prime(''n'')}} − ''b''{{sup|prime(''n'')}}}} is prime are
:1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... {{OEIS|id=A222119}}

==See also==
{{div col|colwidth=25em}}
* [[Repunit]]
* [[Fermat prime]]
* [[Power of two]]
* [[Erdős–Borwein constant]]
* [[Mersenne conjectures]]
* [[Mersenne twister]]
* [[Double Mersenne number]]
* [[Prime95]] / MPrime
* [[Great Internet Mersenne Prime Search]] (GIMPS)
* [[Largest known prime number]]
* [[Titanic prime]]
* [[Gigantic prime]]
* [[Megaprime]]
* [[Wieferich prime]]
* [[Wagstaff prime]]
* [[Cullen prime]]
* [[Woodall prime]]
* [[Proth prime]]
* [[Solinas prime]]
* [[Gillies' conjecture]]
* [[Williams number]]
{{div col end}}

==References==
{{Reflist|30em}}

==External links==
{{Wiktionary|Mersenne prime}}
{{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}}
*{{springer|title=Mersenne number|id=p/m063480}}
*[http://www.mersenne.org GIMPS home page]
*[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 the largest known Mersenne primes
*[http://www.mersenne.org/report_factors/ GIMPS, known factors of Mersenne numbers]
*{{math|''M''<sub>''q''</sub> {{=}} (8''x'')<sup>2</sup> − (3''qy'')<sup>2</sup>}} [http://tony.reix.free.fr/Mersenne/Mersenne8x3qy.pdf Property of Mersenne numbers with prime exponent that are composite] (PDF)
*{{math|''M''<sub>''q''</sub> {{=}} ''x''<sup>2</sup> + ''d''·''y''<sup>2</sup>}} [http://www.math.leidenuniv.nl/scripties/jansen.ps math thesis] (PS)
*{{cite web|last=Grime|first=James|title=31 and Mersenne Primes|url=http://www.numberphile.com/videos/31.html|work=Numberphile|publisher=[[Brady Haran]]}}
*[http://www.utm.edu/research/primes/mersenne/LukeMirror/biblio.htm Mersenne prime bibliography] with hyperlinks to original publications
*[http://www.taz.de/pt/2005/03/11/a0355.nf/text report about Mersenne primes] — detection in detail {{de icon}}
*[http://mersennewiki.org/index.php/Main_Page GIMPS wiki]
*[https://web.archive.org/web/20141014102940/http://www.garlic.com/~wedgingt/mersenne.html Will Edgington's Mersenne Page] — contains factors for small Mersenne numbers
*[http://www.mersenne.org/report_factors/ Known factors of Mersenne numbers]
*[http://www.isthe.com/chongo/tech/math/prime/mersenne.html Decimal digits and English names of Mersenne primes]
*[http://primes.utm.edu/curios/page.php/2305843009213693951.html Prime curios: 2305843009213693951]
*[http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt Factorization of Mersenne numbers {{math|''M<sub>n</sub>''}}, with {{math|''n''}} odd, {{math|''n''}} up to 1199]
*[http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt Factorization of Mersenne numbers {{math|''M''<sub>2''n''</sub>}}, {{math|2''n''}} up to 2398 ({{math|''n''}} up to 1199) or {{math|2''n''}} is in the form {{math|8''k'' + 4}} up to 4796 ({{math|''n''}} is on the form {{math|4''k'' + 2}} up to 2398)]
*{{OEIS el|1=A250197|2=Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime}}—Factorization of Mersenne numbers {{math|''M<sub>n</sub>''}} ({{math|''n''}} up to 1280)
*[https://web.archive.org/web/20141015022019/http://www.garlic.com/~wedgingt/factoredM.txt Factorization of completely factored Mersenne numbers]
*[https://homes.cerias.purdue.edu/~ssw/cun/index.html The Cunningham project, factorization of {{math|1=''b<sup>n</sup>'' ± 1, ''b'' = 2, 3, 5, 6, 7, 10, 11, 12}}]
*[http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm Factorization of {{math|''b<sup>n</sup>'' ± 1, 2 ≤ ''b'' ≤ 12}}]
*[http://www.leyland.vispa.com/numth/factorization/anbn/main.htm Factorization of {{math|''a<sup>n</sup>'' ± ''b<sup>n</sup>''}}, with coprime {{math|''a'', ''b'', 2 ≤ ''b'' < ''a'' ≤ 12}}]

===MathWorld links===
* {{mathworld|urlname=MersenneNumber|title=Mersenne number}}
* {{mathworld|urlname=MersennePrime|title=Mersenne prime}}
* [http://mathworld.wolfram.com/news/2009-06-07/mersenne-47/ 47th Mersenne Prime Found]

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers}}
{{Mersenne}}
{{Large numbers}}

{{DEFAULTSORT:Mersenne Prime}}
[[Category:Articles containing proofs]]
[[Category:Classes of prime numbers]]
[[Category:Unsolved problems in mathematics]]
[[Category:Great Internet Mersenne Prime Search]]
[[Category:Integer sequences]]