Editing Great Internet Mersenne Prime Search

{{Short description|Volunteer project using software to search for Mersenne prime numbers}}
{{Distinguish|GIMP}}
{{Infobox distributed computing project
| name                   = Great Internet Mersenne Prime Search (GIMPS)
| logo                   = GIMPS logo 2020.png
| logo_caption           = Logo
| software               = [[prime95]]
| total users            = 280,000
| total hosts            = 2,900,000<ref>{{cite web |title=PrimeNet Statistics |url=https://www.mersenne.org/primenet/ |website=www.mersenne.org |access-date=6 April 2025}}</ref>
| website                = {{Official URL}}
}}
The '''Great Internet Mersenne Prime Search''' ('''GIMPS''') is a collaborative project of volunteers who use freely available [[software]] to search for [[Mersenne prime]] numbers.

GIMPS was founded in 1996 by [[George Woltman]], who also wrote the [[Prime95]] client and its Linux port MPrime. Scott Kurowski wrote the back-end PrimeNet [[server (computing)|server]] to demonstrate volunteer computing software by Entropia, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. with Kurowski as Executive Vice President and board director. GIMPS is said to be one of the first large-scale [[volunteer computing]] projects over the Internet for research purposes.<ref>{{cite web|title=Volunteer computing|url=https://boinc.berkeley.edu/trac/wiki/VolunteerComputing|publisher=BOINC|access-date=25 December 2021|url-status=live|archive-url=https://web.archive.org/web/20211218111656/https://boinc.berkeley.edu/trac/wiki/VolunteerComputing|archive-date=18 December 2021}}</ref>

{{As of|2024|10}}, the project has found a total of eighteen Mersenne primes, sixteen of which were the [[largest known prime number]] at their respective times of discovery. The largest known [[prime number|prime]] {{as of|2024|10|lc=on|url=http://primes.utm.edu/top20/page.php?id=3}} is 2<sup>136,279,841</sup>&nbsp;−&nbsp;1 (or M<sub>136,279,841</sub> for short) and was discovered on October 12, 2024, by Luke Durant.<ref name="GIMPS-2024">{{cite web |title=GIMPS Discovers Largest Known Prime Number: 2<sup>136,279,841</sup> − 1 |url=https://www.mersenne.org/primes/?press=M136279841 |date=21 October 2024 |work=Mersenne Research, Inc. |access-date=21 October 2024}}</ref><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. |access-date=21 December 2018 }}</ref> On December 4, 2020, the project passed a major milestone after all exponents below 100 million were checked at least once.<ref>{{cite web |title=GIMPS Milestones Report |url=http://www.mersenne.org/report_milestones/ |website=Mersenne.org |publisher=Mersenne Research, Inc. |access-date=5 December 2020}}</ref>

From its inception until 2018, the project relied primarily on the [[Lucas–Lehmer primality test]]<ref>[http://www.mersenne.org/faq.htm#what ''What are Mersenne primes? How are they useful?''] - GIMPS Home Page</ref> as it is an [[algorithm]] that is both specialized for testing Mersenne primes and particularly efficient on [[Binary numeral system|binary]] [[computer architecture]]s. Before applying it to a given Mersenne number, there was a [[trial division]] phase, used to rapidly eliminate many Mersenne numbers with small factors. [[Pollard's p − 1 algorithm|Pollard's ''p'' − 1 algorithm]] is also used to search for [[smooth number|smooth]] factors.

In 2018, GIMPS adopted a [[Fermat primality test]] with basis a=3<ref>a=2 wouldn't work as all Mersenne numbers are 2-pseudoprimes.</ref><ref>https://www.mersenneforum.org/node/22795</ref>as an alternative option for primality testing,<ref>{{cite web | url=https://www.mersenne.org/various/math.php#lucas-lehmer | title=GIMPS - the Math - PrimeNet }}</ref> while keeping the Lucas–Lehmer test as a double-check for Mersenne numbers detected as [[probable prime]]s by the Fermat test.<ref>{{Cite web |title=mersenneforum.org - View Single Post - Getting reliable LL from unreliable hardware |url=https://mersenneforum.org/showpost.php?p=465743&postcount=116 |access-date=2022-10-05 |website=mersenneforum.org}}</ref> (While the Lucas–Lehmer test is deterministic and the Fermat test is only probabilistic, the probability of the Fermat test finding a [[Fermat pseudoprime]] that is not prime is vastly lower than the error rate of the Lucas–Lehmer test due to [[Soft error|computer hardware errors]].<ref>{{Cite web |title=mersenneforum.org - View Single Post - Getting reliable LL from unreliable hardware |url=https://mersenneforum.org/showpost.php?p=465798&postcount=127 |access-date=2022-10-05 |website=mersenneforum.org}}</ref>{{bettersource||Can this forum post be backed up by a mathematical publication? The article [[Fermat pseudoprime]] gives a sourced estimate for the probability of a composite pseudoprime for random base b, but not for fixed base 3|date=April 2025}})

In September 2020,<ref>{{cite web |title=Announcements |url=https://www.mersenne.org/ |publisher=GIMPS, the Great Internet Mersenne Prime Search |archive-url=https://web.archive.org/web/20210814062113/https://www.mersenne.org/ |access-date=1 September 2021|archive-date=2021-08-14 }}</ref><ref>{{cite web |title=What's new |url=https://www.mersenne.org/download/whatsnew_303b6.txt |access-date=1 September 2021}}</ref><ref>{{cite web |title=Prime95 v30.3 |url=https://www.mersenneforum.org/showthread.php?t=25823 |access-date=1 September 2021}}</ref> GIMPS began to support [[primality certificate|primality proofs]] based on verifiable delay functions.<ref>{{cite web|url=https://mersenneforum.org/showthread.php?t=25638|title=The Next Big Development for GIMPS|last=Woltman|first=George|date=2020-06-16|work=GIMPS forum|access-date=20 May 2022}}</ref> The proof files are generated while the Fermat primality test is in progress. These proofs, together with an error-checking algorithm devised by Robert Gerbicz, provide a complete confidence in the correctness of the test result and eliminate the need for double checks. First-time Lucas–Lehmer tests were deprecated in April 2021.<ref>{{cite web|url=https://mersenneforum.org/showthread.php?p=575517|title=First time LL is no more|last=Woltman|first=George|date=2021-04-08|access-date=19 May 2022}}</ref>

GIMPS also has sub-projects to factor known composite Mersenne and [[Fermat number]]s.<ref>{{cite web|url=https://mersenne.org/report_ecm|title=PrimeNet ECM Progress|access-date=20 May 2022}}</ref>

==History==
The project began in early January 1996,<ref>[http://www.garlic.com/~wedgingt/newsletters.html#9 The Mersenne Newsletter, Issue #9. Retrieved 2011-10-02.] {{webarchive|url=https://web.archive.org/web/20120206050420/http://www.garlic.com/~wedgingt/newsletters.html |date=2012-02-06 }}</ref><ref>{{cite web|url=https://www.mersenneforum.org/showpost.php?p=69824&postcount=3|title=mersenneforum.org - View Single Post - Party on! GIMPS turns 10!!!|website=www.mersenneforum.org|access-date=22 December 2018}}</ref> with a program that ran on [[Intel 80386|i386]] computers.<ref>{{cite web
 | url= http://www.mersenne.org/newsletters/news1.txt
 | title= The Mersenne Newsletter, issue #1
 | last= Woltman | first= George | author-link= George Woltman | date= February 24, 1996
 | format= txt | publisher= Great Internet Mersenne Prime Search (GIMPS)
 | access-date= 2009-06-16 }}</ref><ref name="news9">{{cite web
 | url= http://www.mersenne.org/newsletters/news9.txt
 | title= The Mersenne Newsletter, issue #9
 | last= Woltman | first= George | date= January 15, 1997 | format= txt | publisher= GIMPS
 | access-date= 2009-06-16 }}</ref>
The name for the project was coined by Luke Welsh, one of its earlier searchers and the co-discoverer of the 29th Mersenne prime.<ref>[http://www.mersenne.org/newsletters/news9.txt The Mersenne Newsletter, Issue #9]. Retrieved 2009-08-25.</ref>
Within a few months, several dozen people had joined, and over a thousand by the end of the first year.<ref name="news9" /><ref>{{cite web
 | url= http://www.mersenne.org/newsletters/news3.txt
 | title= The Mersenne Newsletter, issue #3
 | last= Woltman | first= George | date= April 12, 1996 | format= txt | publisher= GIMPS
 | access-date= 2009-06-16 }}</ref>
Joel Armengaud, a participant, discovered the primality of M<sub>1,398,269</sub> on November&nbsp;13,&thinsp;1996.<ref>{{cite web
 | url= http://www.mersenne.org/newsletters/news8.txt
 | title= The Mersenne Newsletter, issue #8
 | last= Woltman | first= George | date= November 23, 1996 | format= txt | publisher= GIMPS
 | access-date= 2009-06-16 }}</ref>
Since then, GIMPS has discovered a new Mersenne prime every 1 to 2 years on average. However, the most recent largest prime found in October 2024 took nearly six years to find.

==Status==
{{As of|2022|07}}, GIMPS has a sustained average aggregate [[throughput]] of approximately 4.71&nbsp;[[FLOPS|PetaFLOPS (or PFLOPS)]].<ref>{{Citation
| url = http://www.mersenne.org/primenet/
| title = PrimeNet Activity Summary
| publisher = GIMPS
| access-date = 2022-07-19
}}</ref> In November 2012, GIMPS maintained 95&nbsp;TFLOPS,<ref>{{Citation
| url = http://www.mersenne.org/primenet/
| title = PrimeNet Activity Summary
| publisher = GIMPS
| access-date = 2012-04-05
}}</ref> theoretically earning the GIMPS [[virtual machine|virtual computer]] a rank of 330 among the [[TOP500]] most powerful known computer systems in the world.<ref>{{cite web|title=TOP500 - November 2012|url=http://www.top500.org/list/2012/11/?page=4|access-date=22 November 2012|archive-date=5 October 2018|archive-url=https://web.archive.org/web/20181005073755/https://www.top500.org/list/2012/11/?page=4|url-status=dead}}</ref> The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of [[Hewlett-Packard]].<ref>TOP500 per November 2012; HP BL460c with 95.1 TFLOP/s (R max).{{cite web|title=TOP500 - Rank 329|url=http://www.top500.org/system/177960|access-date=22 November 2012}}</ref> As of July 2021 TOP500 results, the current GIMPS numbers would no longer make the list.

Previously, this was approximately 50&nbsp;TFLOPS in early 2010, 30&nbsp;TFLOPS in mid-2008, 20&nbsp;TFLOPS in mid-2006, and 14&nbsp;TFLOPS in early 2004.

==Software license==
Although the GIMPS software's [[source code]] is publicly available,<ref>{{cite web | url=http://www.mersenne.org/freesoft/default.php#source | title=Software Source Code | publisher=Mersenne Research, Inc. | access-date=March 16, 2013}}</ref> technically it is not [[free software]], since it has a restriction that users must abide by the project's distribution terms.<ref name="legal">{{Citation
| url = http://www.mersenne.org/legal/
| title = GIMPS Legalese
| publisher = GIMPS
| access-date = 2011-09-19
}}</ref>
Specifically, if the software is used to discover a prime number with at least 100,000,000 decimal digits, the user will only win $50,000 of the $150,000 prize offered by the [[Electronic Frontier Foundation]]. On the other hand, they will win $3,000 when discovering a smaller prime not qualifying for the prize.<ref name="legal" /><ref>{{Citation
| url = https://www.eff.org/awards/coop
| title = EFF Cooperative Computing Awards
| date = 29 February 2008
| publisher = Electronic Frontier Foundation
| access-date = 2011-09-19
}}</ref>

Third-party programs for testing Mersenne numbers, such as Mlucas<ref>{{Cite web|url=https://www.mersenneforum.org/mayer/README.html|title = Mlucas README}}</ref> and Glucas<ref>{{Cite web|url=http://glucas.sourceforge.net/glucas/|title = Untitled}}</ref> (for non-x86 systems), do not have this restriction.

GIMPS also "reserves the right to change this [[End-user license agreement|EULA]] without notice and with reasonable retroactive effect''.''"<ref name="legal" />

==Primes found==
All Mersenne primes are of the form {{nowrap|M<sub>''p''</sub> {{=}} 2<sup>''p''</sup> − 1}}, where ''p'' is a prime number itself. The smallest Mersenne prime in this table is {{nowrap|2<sup>1398269</sup> − 1.}}

The first column is the rank of the Mersenne prime in the (ordered) [[integer sequence|sequence]] of all Mersenne primes;<ref>{{cite web|title=GIMPS List of Known Mersenne Prime Numbers|url=https://www.mersenne.org/primes/|publisher=Mersenne Research, Inc.|access-date=2018-01-03}}</ref> GIMPS has found all known Mersenne primes beginning with the 35th.
{| class="wikitable sortable"
|-
! # || Discovery date || Prime M<sub>''p''</sub> || Digits count || Processor
|-
| style="text-align:center;"| 35 || November 13, 1996 || style="text-align:left;"| M<sub>1398269</sub> || style="text-align:right;"| 420,921 || [[Pentium]] (90 [[Hertz|MHz]])
|-
| style="text-align:center;"| 36 || August 24, 1997 || style="text-align:left;"| M<sub>2976221</sub> || style="text-align:right;"| 895,932 || Pentium (100&nbsp;MHz)
|-
| style="text-align:center;"| 37 || January 27, 1998 || style="text-align:left;"| M<sub>3021377</sub> || style="text-align:right;"| 909,526 || Pentium (200&nbsp;MHz)
|-
| style="text-align:center;"| 38 || June 1, 1999 || style="text-align:left;"| M<sub>6972593</sub> || style="text-align:right;"| 2,098,960 || Pentium (350&nbsp;MHz)
|-
| style="text-align:center;"| 39 || November 14, 2001 || style="text-align:left;"| M<sub>13466917</sub> || style="text-align:right;"| 4,053,946 || [[Advanced Micro Devices|AMD]] [[Athlon Thunderbird|T-Bird]] (800&nbsp;MHz)
|-
| style="text-align:center;"| 40 || November 17, 2003 || style="text-align:left;"| M<sub>20996011</sub> || style="text-align:right;"| 6,320,430 || Pentium (2&nbsp;GHz)
|-
| style="text-align:center;"| 41 || May 15, 2004 || style="text-align:left;"| M<sub>24036583</sub> || style="text-align:right;"| 7,235,733 || [[Pentium 4]] (2.4&nbsp;GHz)
|-
| style="text-align:center;"| 42 || February 18, 2005 || style="text-align:left;"| M<sub>25964951</sub> || style="text-align:right;"| 7,816,230 || Pentium 4 (2.4&nbsp;GHz)
|-
| style="text-align:center;"| 43 || December 15, 2005 || style="text-align:left;"| M<sub>30402457</sub> || style="text-align:right;"| 9,152,052 || Pentium 4 (2&nbsp;GHz [[Overclocking|overclocked]] to 3&nbsp;GHz)
|-
| style="text-align:center;"| 44 || September 4, 2006 || style="text-align:left;"| M<sub>32582657</sub> || style="text-align:right;"| 9,808,358 || Pentium 4 (3&nbsp;GHz)
|-
| style="text-align:center;"| 45 || September 6, 2008 || style="text-align:left;"| M<sub>37156667</sub> || style="text-align:right;"| 11,185,272 || Intel [[Core 2 Duo]] (2.83&nbsp;GHz)
|-
| style="text-align:center;"| 46 || June 4, 2009 || style="text-align:left;"| M<sub>42643801</sub> || style="text-align:right;"| 12,837,064 || Intel Core 2 Duo (3&nbsp;GHz)
|-
| style="text-align:center;"| 47 || August 23, 2008 || style="text-align:left;"| M<sub>43112609</sub> || style="text-align:right;"| 12,978,189 || Intel Core 2 Duo E6600 CPU (2.4&nbsp;GHz)
|-
| style="text-align:center;"| 48 || January 25, 2013 || style="text-align:left;"| M<sub>57885161</sub> || style="text-align:right;"| 17,425,170 || Intel Core 2 Duo E8400 @ 3.00&nbsp;GHz
|-
| style="text-align:center;"| 49{{ref label|unverified_index|†|^ †}} || January 7, 2016 || style="text-align:left;"| M<sub>74207281</sub> || style="text-align:right;"| 22,338,618 || Intel [[Haswell (microarchitecture)|Core i7-4790]]
|-
| style="text-align:center;"| 50{{ref label|unverified_index|†|^ †}} || December 26, 2017 || style="text-align:left;"| M<sub>77232917</sub> || style="text-align:right;"| 23,249,425 || Intel [[Skylake (microarchitecture)|Core i5-6600]]
|-
| style="text-align:center;"| 51{{ref label|unverified_index|†|^ †}} || December 7, 2018 || style="text-align:left;"| M<sub>82589933</sub> || style="text-align:right;"| 24,862,048 || Intel Core i5-4590T
|-
| style="text-align:center;"| 52{{ref label|unverified_index|†|^ †}} || October 21, 2024 || style="text-align:left;"| M<sub>136279841</sub>{{ref label|number_size|‡|^ ‡}} || style="text-align:right;"| 41,024,320 || [[Nvidia A100]]
|-
|}

{{note label|unverified_index|†|^ †}} {{As of|2025|5|14|df=US}}, 73,546,481 is the largest exponent below which all other prime exponents have been checked twice, so it is not verified whether any undiscovered Mersenne primes exist between the 48th (M<sub>57885161</sub>) and the 52nd (M<sub>136279841</sub>) on this chart; the ranking is therefore provisional. Furthermore, 130,439,863 is the largest exponent below which all other prime exponents have been tested at least once, so all Mersenne numbers below the 51st (M<sub>82589933</sub>) have been tested.<ref>{{cite web|title=GIMPS Milestones|url=http://www.mersenne.org/report_milestones/|publisher=Mersenne Research, Inc.|access-date=2020-11-30}}</ref>

{{note label|number_size|‡|^ ‡}} The number M<sub>136279841</sub> has 41,024,320 decimal digits. To help visualize the size of this number, if it were to be saved to disk, the resulting text file would be nearly 42 megabytes long (most books in plain text format are under two megabytes).  A standard [[word processor]] layout (50 lines per page, 75 digits per line) would require 10,940 pages to display it. If one were to print it out using standard printer paper, single-sided, it would require approximately 22 [[Paper ream|reams]] (22 × 500 = 11,000 sheets) of paper.

Whenever a possible prime is reported to the server, it is verified first (by one or more independent tests on different machines) before being announced. The importance of this was illustrated in 2003, when a false positive was reported to the server as being a Mersenne prime but verification failed.<ref>{{cite web|url=https://mersenneforum.org/showthread.php?p=6149|title=M40, what went wrong? - Page 11 - mersenneforum.org|website=mersenneforum.org|access-date=22 December 2018}}</ref>

The official "discovery date" of a prime is the date that a human first noticed the result for the prime, which may differ from the date that the result was first reported to the server. For example, M<sub>74207281</sub> was reported to the server on September 17, 2015, but the report was overlooked until January 7, 2016.<ref>{{cite web|url=https://www.mersenne.org/primes/?press=M74207281|title=GIMPS Project Discovers Largest Known Prime Number|date=January 19, 2016}}</ref>

==See also==
* [[Berkeley Open Infrastructure for Network Computing]]
* [[List of volunteer computing projects]]
* [[PrimeGrid]]

==References==
{{Reflist}}

==External links==
* {{Official website|www.mersenne.org}}

{{Mersenne}}

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[[Category:Great Internet Mersenne Prime Search| ]]
[[Category:Distributed prime searches]]
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[[Category:Volunteer computing projects]]