The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available

software Software consists of computer programs that instruct the Execution (computing), execution of a computer. Software also includes design documents and specifications. The history of software is closely tied to the development of digital comput ...

to search for

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...

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 Volunteer computing is a type of distributed computing in which people donate their computers' unused resources to a research-oriented project, and sometimes in exchange for credit points. The fundamental idea behind it is that a modern desktop ...

projects over the Internet for research purposes. , the project has found a total of eighteen Mersenne primes, sixteen of which were the

largest known prime number The largest known prime number is , a number which has 41,024,320 digits when written in the decimal system. It was found on October 12, 2024, on a cloud-based virtual machine volunteered by Luke Durant, a 36-year-old researcher from San Jose, Cali ...

at their respective times of discovery. The largest known

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

is 2^136,279,841 − 1 (or M_136,279,841 for short) and was discovered on October 12, 2024, by Luke Durant. On December 4, 2020, the project passed a major milestone after all exponents below 100 million were checked at least once. From its inception until 2018, the project relied primarily on the Lucas–Lehmer primality test as it is an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

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. In 2018, GIMPS adopted a Fermat primality test with basis a=3as an alternative option for primality testing, while keeping the Lucas–Lehmer test as a double-check for Mersenne numbers detected as

probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific co ...

s by the Fermat test. (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 computer hardware errors.) In September 2020, GIMPS began to support primality proofs based on verifiable delay functions. 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. GIMPS also has sub-projects to factor known composite Mersenne and

Fermat number In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a natural number, positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers ...

History

The project began in early January 1996, with a program that ran on

i386 The Intel 386, originally released as the 80386 and later renamed i386, is the third-generation x86 architecture microprocessor from Intel. It was the first 32-bit processor in the line, making it a significant evolution in the x86 archite ...

computers. The name for the project was coined by Luke Welsh, one of its earlier searchers and the co-discoverer of the 29th Mersenne prime. Within a few months, several dozen people had joined, and over a thousand by the end of the first year. Joel Armengaud, a participant, discovered the primality of M_1,398,269 on November 13, 1996. 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

, GIMPS has a sustained average aggregate

throughput Network throughput (or just throughput, when in context) refers to the rate of message delivery over a communication channel in a communication network, such as Ethernet or packet radio. The data that these messages contain may be delivered ov ...

of approximately 4.71 PetaFLOPS (or PFLOPS). In November 2012, GIMPS maintained 95 TFLOPS, theoretically earning the GIMPS virtual computer a rank of 330 among the

TOP500 The TOP500 project ranks and details the 500 most powerful non-distributed computing, distributed computer systems in the world. The project was started in 1993 and publishes an updated list of the supercomputers twice a year. The first of these ...

most powerful known computer systems in the world. The preceding place was then held by an 'HP Cluster Platform 3000 BL460c G7' of

Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company. It was founded by Bill Hewlett and David Packard in 1939 in a one-car garage in Palo Alto, California ...

. 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 license

Although the GIMPS software's

source code In computing, source code, or simply code or source, is a plain text computer program written in a programming language. A programmer writes the human readable source code to control the behavior of a computer. Since a computer, at base, only ...

is publicly available, technically it is not

free software Free software, libre software, libreware sometimes known as freedom-respecting software is computer software distributed open-source license, under terms that allow users to run the software for any purpose as well as to study, change, distribut ...

, since it has a restriction that users must abide by the project's distribution terms. 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 The Electronic Frontier Foundation (EFF) is an American international non-profit digital rights group based in San Francisco, California. It was founded in 1990 to promote Internet civil liberties. It provides funds for legal defense in court, ...

. On the other hand, they will win $3,000 when discovering a smaller prime not qualifying for the prize. 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 An end-user license agreement or EULA () is a legal contract between a software supplier and a customer or end-user. The practice of selling licenses to rather than copies of software predates the recognition of software copyright, which has ...

without notice and with reasonable retroactive effect''.''"

Primes found

All Mersenne primes are of the form , where ''p'' is a prime number itself. The smallest Mersenne prime in this table is The first column is the rank of the Mersenne prime in the (ordered)

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of all Mersenne primes; GIMPS has found all known Mersenne primes beginning with the 35th. , 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_57885161) and the 52nd (M_136279841) 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_82589933) have been tested. The number M_136279841 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 A word processor (WP) is a device or computer program that provides for input, editing, formatting, and output of text, often with some additional features. Early word processors were stand-alone devices dedicated to the function, but current word ...

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. 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_74207281 was reported to the server on September 17, 2015, but the report was overlooked until January 7, 2016.

References

External links