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Great Internet Mersenne Prime Search
Volunteer project using software to search for Mersenne prime numbers From Wikipedia, the free encyclopedia
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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 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.[2]
As of October 2024[update], 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 as of October 2024[ref] is 2136,279,841 − 1 (or M136,279,841 for short) and was discovered on October 12, 2024, by Luke Durant,[3][4] and оn June 18, 2025, the project passed a milestone after all exponents below 136,279,841 were checked at least once.[5]
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Algorithm
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From its inception until 2018, the project relied primarily on the Lucas–Lehmer primality test (LL)[6] as it is an algorithm that is both specialized for testing Mersenne primes and particularly efficient on binary computer architectures. 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 is also used to search for smooth factors. The variant of LL used in the main implementation (Prime95) is specifically based on irrational base discrete weighted transform with double-precision floating-point numbers, which provide a efficient way to square a large number modulo 2P-1.[7]
Special care is taken to make sure that the use of floating-point numbers does not introduce errors into the LL calculation. The program verifies that the round-off error is no greater than 0.4 every 128 iterations, or if the exponent being tested is within 0.5% of the maximum exponent size that can be handled by the size of FFT in use (or if requested using a special option), at every single iteration. Every 12 hours the program runs an additional error check based on the Jacobi symbol,[8] with 50% chance of catching an error. On top of these, each completed LL calculation is repeated by a different hardware for "double-checking". Based on historical double-check data, each LL calculation without any serious error reported had an error rate of 1.5%; those with at least one serious error reported had an error rate of 50%.[7]
In 2018, GIMPS adopted a Fermat primality test with basis a=3[a] as an alternative option for primality testing,[10] while keeping the Lucas–Lehmer test as a double-check for Mersenne numbers detected as probable primes by the Fermat test.[11] This new test is called PRP (probable prime) in GIMPS parlance. Using a method devised by Robert Gerbicz, GIMPS can be "99.999+%" sure that a PRP result is generated correctly.[7] As a result, even though 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 computer hardware errors.[12][better source needed]
In September 2020,[13][14][15] GIMPS began to support primality proofs based on verifiable delay functions countributed by Krzysztof Pietrzak.[16] The proof files are generated while the Fermat primality test is in progress. These proofs, together with Gerbicz's error-checking algorithm (see above), provide a complete confidence in the correctness of the test result and eliminate the need for double checks (check of proof could be run in 1/100 of the time of the original Fermat computation).[7] First-time Lucas–Lehmer tests were deprecated in April 2021, leaving LL to only be used on probable primes found by the Fermat test.[17] PRP and LL are very similar in terms of runtime;[18] the preference comes from a higher confidence in PRP results.[17]
GIMPS also has sub-projects to factor known composite Mersenne and Fermat numbers. These use the elliptic-curve factorization method and Williams's p + 1 algorithm.[19][b]
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History
The project began in early January 1996,[20][21] with a program that ran on i386 computers.[22][23] The name for the project was coined by Luke Welsh, one of its earlier searchers and the co-discoverer of the 29th Mersenne prime.[24] Within a few months, several dozen people had joined, and over a thousand by the end of the first year.[23][25] Joel Armengaud, a participant, discovered the primality of M1,398,269 on November 13, 1996.[26] 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.
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Status
As of July 2022[update], GIMPS has a sustained average aggregate throughput of approximately 4.71 PetaFLOPS (or PFLOPS).[27] In November 2012, GIMPS maintained 95 TFLOPS,[28] theoretically earning the GIMPS virtual computer a rank of 330 among the TOP500 most powerful known computer systems in the world.[29] The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of Hewlett-Packard.[30] As of July 2021 TOP500 results, the current 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
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Prime95
The primary software used by GIMPS is Prime95, which implements all algorithms for an x86 or x86-64 CPU: trial factoring (usually left to GPUs), PRP, P-1, P+1, ECM, and PRP certification. Although the Prime95 software's source code is publicly available,[31] technically it is not free software, since it has a restriction that users must abide by the project's distribution terms.[32] 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.[32][33]
GIMPS also "reserves the right to change this EULA without notice and with reasonable retroactive effect."[32]
Third-party software
The third-party software do not share the same restriction as Prime95. They can be used to join GIMPS using a program called AutoPrimeNet, which fetches tasks from GIMPS and send back the results. Available software include:[34]
- Mlucas, which implements LL, Fermat PRP, and Pépin's test. Shipped with MFactor for trial-factoring. Able to run on x86, x86-64, ARM, and most other CPU architectures. Uses double-precision IBDWT.[35]
- Glucas, outdated implementation of LL for x86 and non-x86 CPUs. Uses double-precision IBDWT.
- GPUowl and PRPLL, OpenCL programs for running PRP and LL intended for GPUs. Uses double-precision IBDWT.
- mfaktc (CUDA) / mfakto (OpenCL), programs for GPU trial factoring using 32-bit integer arithmetic.
- CUDALucas, outdated implementation of LL for CUDA. Uses double-precision IBDWT.
- PrMers/Marin, implements LL and PRP. Uses IBDWT with number-theoretic transform over Z/(264 - 232 + 1)Z using 64-bit integer arithmetic.
In addition, PrimeNet accepts other forms of data contributions from projects such as TJOAI (Tadashi Taura's custom software to trial-factor many Mersenne numbers at once).
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Primes found
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All Mersenne primes are of the form Mp = 2p − 1, where p is a prime number itself. The smallest Mersenne prime in this table is 21398269 − 1.
The first column is the rank of the Mersenne prime in the (ordered) sequence of all Mersenne primes;[36] GIMPS has found all known Mersenne primes beginning with the 35th.
^ † As of August 30, 2025[update], 75,553,609 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 49th (M74207281) and the 52nd (M136279841) on this chart; the ranking is therefore provisional. Furthermore, 138,249,913 is the largest exponent below which all other prime exponents have been tested at least once, so all Mersenne numbers below the 52nd have been tested.[37]
^ ‡ The number M136279841 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 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.[38]
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, M74207281 was reported to the server on September 17, 2015, but the report was overlooked until January 7, 2016.[39]
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See also
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External links
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