Editing Prime number (section)

=== Special-purpose algorithms and the largest known prime ===
{{Further|List of prime numbers}}
In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the [[Lucas–Lehmer primality test]] can determine whether a [[Mersenne number]] (one less than a [[power of two]]) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.<ref>{{cite book | last = Tao | first = Terence | author-link = Terence Tao | contribution = 1.7 The Lucas–Lehmer test for Mersenne primes | contribution-url = https://terrytao.wordpress.com/2008/10/02/the-lucas-lehmer-test-for-mersenne-primes/ | isbn = 978-0-8218-4883-8 | location = Providence, RI | mr = 2523047 | pages = 36–41 | publisher = American Mathematical Society | title = Poincaré's legacies, pages from year two of a mathematical blog. Part I | year = 2009 }}</ref> This is why since 1992 ({{as of|2024|10|lc=y}}) the [[largest known prime|largest ''known'' prime]] has always been a Mersenne prime.<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA41 p.&nbsp;41].</ref> It is conjectured that there are infinitely many Mersenne primes.<ref>For instance see {{harvnb|Guy|2013}}, [https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA13 A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape {{tmath| k\cdot 2^n+1 }}. pp. 13–21].</ref>

The following table gives the largest known primes of various types. Some of these primes have been found using [[distributed computing]]. In 2009, the [[Great Internet Mersenne Prime Search]] project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits.<ref>{{cite web | url= https://www.eff.org/press/archives/2009/10/14-0  | title= Record 12-Million-Digit Prime Number Nets $100,000 Prize | date= October 14, 2009 | publisher= Electronic Frontier Foundation | access-date= 2010-01-04 }}</ref> The [[Electronic Frontier Foundation]] also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.<ref>{{cite web  | url= https://www.eff.org/awards/coop | title= EFF Cooperative Computing Awards | date= 2008-02-29| publisher= Electronic Frontier Foundation | access-date= 2010-01-04 }}</ref>

{| class="wikitable"
|-
! Type
! Prime
! Number of decimal digits
! Date
! Found by
|-
| [[Mersenne prime]]
| 2<sup>136,279,841</sup> − 1
| style="text-align:right;"| 41,024,320
| October 21, 2024<ref name="GIMPS-2024"/>
| Luke Durant, [[Great Internet Mersenne Prime Search]]
|-
| [[Proth prime]]
| 10,223 × 2<sup>31,172,165</sup> + 1
| style="text-align:right;"| 9,383,761
| October 31, 2016<ref>{{cite web|url=https://www.primegrid.com/download/SOB-31172165.pdf|title=PrimeGrid's Seventeen or Bust Subproject|access-date=2017-01-03}}</ref>
| Péter Szabolcs, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=3 |title=The Top Twenty: Largest Known Primes |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[factorial prime]]
| 208,003! − 1
| style="text-align:right;"| 1,015,843
| July 2016
| Sou Fukui<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=30 |title=The Top Twenty: Factorial |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[primorial prime]]{{efn|The [[primorial]] function of {{tmath|n}}, denoted by {{tmath|n\#}}, yields the product of the prime numbers up to {{tmath|n}}, and a [[primorial prime]] is a prime of one of the forms {{tmath| n\#\pm 1 }}.<ref>{{harvnb|Ribenboim|2004}}, p. 4.</ref>}}
| 1,098,133# − 1
| style="text-align:right;"| 476,311
| March 2012
| James P. Burt, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=5 |title=The Top Twenty: Primorial |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[twin prime]]s
| 2,996,863,034,895 × 2<sup>1,290,000</sup> ± 1
| style="text-align:right;"| 388,342
| September 2016
| Tom Greer, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=1 |title=The Top Twenty: Twin Primes |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|}