# Great Internet Mersenne Prime Search

The **Great Internet Mersenne Prime Search** (**GIMPS**) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers.

The GIMPS project was founded by George Woltman in 1996, who also wrote the software Prime95 and MPrime for the project. Scott Kurowski wrote the PrimeNet Internet server that supports the research to demonstrate Entropia-distributed computing software, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. Kurowski is Executive Vice President and board director of Mersenne Research Inc. GIMPS is said to be one of the first large scale distributed computing projects over the Internet for research purposes.^{[1]}

As of December 2018^{[update]}, the project has found a total of seventeen Mersenne primes, fifteen of which were the largest known prime number at their respective times of discovery. The largest known prime as of December 2018^{[ref]} is 2^{82,589,933} − 1 (or M_{82,589,933} in short). This prime was discovered on December 7, 2018 by Patrick Laroche.^{[2]}

To perform its testing, the project relies primarily on Lucas–Lehmer primality test,^{[3]} an algorithm that is both specialized to testing Mersenne primes and particularly efficient on binary computer architectures. They also have a trial division phase, used to rapidly eliminate Mersenne numbers with small factors which make up a large proportion of candidates. Pollard's p − 1 algorithm is also used to search for larger factors.

## Contents

## History[edit]

The project began in early January 1996,^{[4]}^{[5]} with a program that ran on i386 computers.^{[6]}^{[7]}
The name for the project was coined by Luther Welsh, one of its earlier searchers and the co-discoverer of the 29th Mersenne prime.^{[8]}
Within a few months, several dozen people had joined, and over a thousand by the end of the first year.^{[7]}^{[9]}
Joel Armengaud, a participant, discovered the primality of M_{1,398,269} on November 13, 1996.^{[10]}

## Status[edit]

As of October 2017^{[update]}, GIMPS has a sustained aggregate throughput of approximately 324 TeraFLOPS (or TFLOPS).^{[11]} In November 2012, GIMPS maintained 95 TFLOPS,^{[12]} theoretically earning the GIMPS virtual computer a rank of 330 among the TOP500 most powerful known computer systems in the world.^{[13]} The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of Hewlett-Packard.^{[14]} As of November 2014 TOP500 results, these old GIMPS numbers would no longer make the list.

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

## Software license[edit]

Although the GIMPS software's source code is publicly available,^{[15]} technically it is not free software, since it has a restriction that users must abide by the project's distribution terms.^{[16]}
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.^{[16]}^{[17]}

Third-party programs for testing Mersenne numbers, such as Mlucas and Glucas (for non-x86 systems), do not have this restriction.

GIMPS also "reserves the right to change this EULA without notice and with reasonable retroactive effect*.*"^{[16]}

## Primes found[edit]

All Mersenne primes are of the form M_{p} = 2^{p} − 1, where *p* is a prime number itself. The smallest Mersenne prime in this table is 2^{1398269} − 1.

The first column is the rank of the Mersenne prime in the (ordered) sequence of all Mersenne primes;^{[18]} GIMPS has found all known Mersenne primes beginning with the 35th.

# | Discovery date | Prime M_{p} |
Digits count | Processor |
---|---|---|---|---|

35 | November 13, 1996 | M_{1398269} |
420,921 | Pentium (90 MHz) |

36 | August 24, 1997 | M_{2976221} |
895,932 | Pentium (100 MHz) |

37 | January 27, 1998 | M_{3021377} |
909,526 | Pentium (200 MHz) |

38 | June 1, 1999 | M_{6972593} |
2,098,960 | Pentium (350 MHz) |

39 | November 14, 2001 | M_{13466917} |
4,053,946 | AMD T-Bird (800 MHz) |

40 | November 17, 2003 | M_{20996011} |
6,320,430 | Pentium (2 GHz) |

41 | May 15, 2004 | M_{24036583} |
7,235,733 | Pentium 4 (2.4 GHz) |

42 | February 18, 2005 | M_{25964951} |
7,816,230 | Pentium 4 (2.4 GHz) |

43 | December 15, 2005 | M_{30402457} |
9,152,052 | Pentium 4 (2 GHz overclocked to 3 GHz) |

44 | September 4, 2006 | M_{32582657} |
9,808,358 | Pentium 4 (3 GHz) |

45 | September 6, 2008 | M_{37156667} |
11,185,272 | Intel Core 2 Duo (2.83 GHz) |

46 | April 12, 2009 | M_{42643801} |
12,837,064 | Intel Core 2 Duo (3 GHz) |

47 | August 23, 2008 | M_{43112609} |
12,978,189 | Intel Core 2 Duo E6600 CPU (2.4 GHz) |

48^{[†]} |
January 25, 2013 | M_{57885161} |
17,425,170 | Intel Core 2 Duo E8400 @ 3.00 GHz |

49^{[†]} |
January 7, 2016 | M_{74207281} |
22,338,618 | Intel Core i7-4790 |

50^{[†]} |
December 26, 2017 | M_{77232917} |
23,249,425 | Intel Core i5-6600 |

51^{[†]} |
December 7, 2018 | M_{82589933}^{[‡]} |
24,862,048 | Intel Core i5-4590T |

** ^{^ †}** As of April 17, 2019

^{[update]}, 47,183,111 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 47th (

*M*

_{43112609}) and the 51st (

*M*

_{82589933}) on this chart; the ranking is therefore provisional. Furthermore, 83,087,503 is the largest exponent below which all other prime exponents have been tested at least once, so all Mersenne numbers below the 51st (

*M*

_{82589933}) have been tested.

^{[19]}

** ^{^ ‡}** The number M

_{82589933}has 24,862,048 decimal digits. To help visualize the size of this number, a standard word processor layout (50 lines per page, 75 digits per line) would require 6,629 pages to display it. If one were to print it out using standard printer paper, single-sided, it would require approximately 14 reams of paper.

Whenever a possible prime is reported to the server, it is verified first before it is announced. The importance of this was illustrated in 2003, when a false positive was reported to possibly be the 40th Mersenne prime but verification failed.^{[20]}

## See also[edit]

- Berkeley Open Infrastructure for Network Computing
- Distributed computing
- List of distributed computing projects
- PrimeGrid

## References[edit]

**^**"Volunteer computing". BOINC. Retrieved 8 October 2012.**^**"GIMPS Project Discovers Largest Known Prime Number: 2^{82,589,933}-1".*Mersenne Research, Inc*. 21 December 2018. Retrieved 21 December 2018.**^***What are Mersenne primes? How are they useful?*- GIMPS Home Page**^**The Mersenne Newsletter, Issue #9. Retrieved 2011-10-02. Archived 2012-02-06 at the Wayback Machine**^**"mersenneforum.org - View Single Post - Party on! GIMPS turns 10!!!".*www.mersenneforum.org*. Retrieved 22 December 2018.**^**Woltman, George (February 24, 1996). "The Mersenne Newsletter, issue #1" (txt). Great Internet Mersenne Prime Search (GIMPS). Retrieved 2009-06-16.- ^
^{a}^{b}Woltman, George (January 15, 1997). "The Mersenne Newsletter, issue #9" (txt). GIMPS. Retrieved 2009-06-16. **^**The Mersenne Newsletter, Issue #9. Retrieved 2009-08-25.**^**Woltman, George (April 12, 1996). "The Mersenne Newsletter, issue #3" (txt). GIMPS. Retrieved 2009-06-16.**^**Woltman, George (November 23, 1996). "The Mersenne Newsletter, issue #8" (txt). GIMPS. Retrieved 2009-06-16.**^***PrimeNet Activity Summary*, GIMPS, retrieved 2017-10-07**^***PrimeNet Activity Summary*, GIMPS, retrieved 2012-04-05**^**"TOP500 - November 2012". Retrieved 22 November 2012.**^**TOP500 per November 2012; HP BL460c with 95.1 TFLOP/s (R max)."TOP500 - Rank 329". Retrieved 22 November 2012.**^**"Software Source Code". Mersenne Research, Inc. Retrieved March 16, 2013.- ^
^{a}^{b}^{c}*GIMPS Legalese*, GIMPS, retrieved 2011-09-19 **^***EFF Cooperative Computing Awards*, Electronic Frontier Foundation, retrieved 2011-09-19**^**"GIMPS List of Known Mersenne Prime Numbers". Mersenne Research, Inc. Retrieved 2018-01-03.**^**"GIMPS Milestones". Mersenne Research, Inc. Retrieved 2019-04-17.**^**"M40, what went wrong? - Page 11 - mersenneforum.org".*mersenneforum.org*. Retrieved 22 December 2018.