Great Internet Mersenne Prime Search

Not to be confused with GIMP.
GIMPS logo

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, 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]

The project has found a total of fifteen Mersenne primes as of January 2016, thirteen of which were the largest known prime number at their respective times of discovery. The largest known prime as of January 2016 is 274,207,281 − 1 (or M74,207,281 in short). This prime was discovered on September 17, 2015 by Curtis Cooper at the University of Central Missouri.[2][3]

To perform its testing, the project relies primarily on Lucas–Lehmer primality test,[4] 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.

History

The project began in early January 1996,[5][6] with a program that ran on i386 computers.[7][8] The name for the project was coined by Luther Welsh, one of its earlier searchers and the discoverer of the 29th Mersenne prime.[9] Within a few months, several dozen people had joined, and over a thousand by the end of the first year.[8][10] Joel Armengaud, a participant, discovered the primality of M1,398,269 on November 13, 1996.[11]

Status

As of March 2013, GIMPS has a sustained aggregate throughput of approximately 137.023 TFLOP/s.[12] In November 2012, GIMPS maintained 95 TFLOP/s,[13] theoretically earning the GIMPS virtual computer a place among the TOP500 most powerful known computer systems in the world. Also theoretically, in November 2012, the GIMPS held a rank of 330 in the TOP500.[14] The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of Hewlett-Packard.[15] As of November 2014 TOP500 results, these old GIMPS numbers would no longer make the list.

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

Software license

Although the GIMPS software's source code is publicly available,[16] technically it is not free software, since it has a restriction that users must abide by the project's distribution terms[17] if the software is used to discover a prime number with at least 100 million decimal digits and wins the $150,000 USD bounty offered by the Electronic Frontier Foundation, and a bounty of $250,000 USD for a prime number with at least 1 billion decimal digits.[18]

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

Also, GIMPS "reserves the right to change this EULA without notice and with reasonable retroactive effect."[17]

Primes found

All Mersenne primes are in the form Mq, where q is the (prime) exponent. The prime number itself is 2q − 1, so the smallest prime number in this table is 21398269 − 1.

Mn is the rank of the Mersenne prime based on its exponent.

Name Mn Discovery date Prime Mq Digits count Processor
M35 November 13, 1996 M1398269 420,921 Pentium (90 MHz)
M36 August 24, 1997 M2976221 895,932 Pentium (100 MHz)
M37 January 27, 1998 M3021377 909,526 Pentium (200 MHz)
M38 June 1, 1999 M6972593 2,098,960 Pentium (350 MHz)
M39 November 14, 2001 M13466917 4,053,946 AMD T-Bird (800 MHz)
M40 November 17, 2003 M20996011 6,320,430 Pentium (2 GHz)
M41 May 15, 2004 M24036583 7,235,733 Pentium 4 (2.4 GHz)
M42 February 18, 2005 M25964951 7,816,230 Pentium 4 (2.4 GHz)
M43 December 15, 2005 M30402457 9,152,052 Pentium 4 (2 GHz overclocked to 3 GHz)
M44 September 4, 2006 M32582657 9,808,358 Pentium 4 (3 GHz)
M45 [*] September 6, 2008 M37156667 11,185,272
M46 [*] April 12, 2009 M42643801 12,837,064 Intel Core 2 Duo (3 GHz)
M47 [*] August 23, 2008 M43112609 12,978,189 Intel Core 2 Duo E6600 CPU (2.4 GHz)
M48 [*] January 25, 2013 M57885161 17,425,170 Intel Core 2 Duo E8400 @ 3.00 GHz
M49 [*] January 7, 2016 M74207281 [**] 22,338,618 Intel Core i7-4790

^ * As of April 20, 2016, 35,613,481 is the largest exponent below which all other exponents have been checked twice, so it is not verified whether any undiscovered Mersenne primes exist between the 44th (M32582657) and the 49th (M74207281) on this chart; the ranking is therefore provisional. Furthermore, 63,350,927 is the largest exponent below which all other exponents have been tested at least once, so some Mersenne numbers between the 48th (M57885161) and the 49th (M74207281) have yet to be tested.[19]

^ ** The number M74207281 has 22,338,618 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 5,957 pages to display it. If one were to print it out using standard printer paper, single-sided, it would require approximately 12 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

References

  1. ↑ "Volunteer computing". BOINC. Retrieved 8 October 2012.
  2. ↑ Cooper, Curtis (7 January 2016). "Mersenne Prime Number discovery - 274207281-1 is Prime!". Mersenne Research, Inc. Retrieved 22 January 2016.
  3. ↑ Chang, Kenneth (21 January 2016). "New Biggest Prime Number = 2 to the 74 Mil ... Uh, It’s Big". New York Times. Retrieved 22 January 2016.
  4. ↑ What are Mersenne primes? How are they useful? - GIMPS Home Page
  5. ↑ The Mersenne Newsletter, Issue #9. Retrieved 2011-10-02.
  6. ↑ Mersenne forum Retrieved 2011-10-02
  7. ↑ Woltman, George (February 24, 1996). "The Mersenne Newsletter, issue #1" (txt). Great Internet Mersenne Prime Search (GIMPS). Retrieved 2009-06-16.
  8. 1 2 Woltman, George (January 15, 1997). "The Mersenne Newsletter, issue #9" (txt). GIMPS. Retrieved 2009-06-16.
  9. ↑ The Mersenne Newsletter, Issue #9. Retrieved 2009-08-25.
  10. ↑ Woltman, George (April 12, 1996). "The Mersenne Newsletter, issue #3" (txt). GIMPS. Retrieved 2009-06-16.
  11. ↑ Woltman, George (November 23, 1996). "The Mersenne Newsletter, issue #8" (txt). GIMPS. Retrieved 2009-06-16.
  12. ↑ PrimeNet Activity Summary, GIMPS, retrieved 2013-03-13
  13. ↑ PrimeNet Activity Summary, GIMPS, retrieved 2012-04-05
  14. ↑ "TOP500 - November 2012". Retrieved 22 November 2012.
  15. ↑ TOP500 per November 2012; HP BL460c with 95.1 TFLOP/s (R max)."TOP500 - Rank 329". Retrieved 22 November 2012.
  16. ↑ "Software Source Code". Mersenne Research, Inc. Retrieved March 16, 2013.
  17. ↑ EFF Cooperative Computing Awards, Electronic Frontier Foundation, retrieved 2011-09-19
  18. ↑ "GIMPS Milestones". Mersenne Research, Inc. Retrieved 2016-04-20.
  19. ↑ http://mersenneforum.org/showthread.php?p=6149

External links

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