Prime95, also distributed as the command-line utility mprime for

FreeBSD FreeBSD is a free-software Unix-like operating system descended from the Berkeley Software Distribution (BSD). The first version was released in 1993 developed from 386BSD, one of the first fully functional and free Unix clones on affordable ...

and

Linux Linux ( ) is a family of open source Unix-like operating systems based on the Linux kernel, an kernel (operating system), operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically package manager, pac ...

, is a

freeware Freeware is software, often proprietary, that is distributed at no monetary cost to the end user. There is no agreed-upon set of rights, license, or EULA that defines ''freeware'' unambiguously; every publisher defines its own rules for the free ...

application Application may refer to: Mathematics and computing * Application software, computer software designed to help the user to perform specific tasks ** Application layer, an abstraction layer that specifies protocols and interface methods used in a ...

written by

George Woltman George Woltman (born November 10, 1957) is the founder of the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project researching Mersenne prime numbers using his software Prime95. He graduated from the Massachusetts Instit ...

. It is the official client of the

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. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...

(GIMPS), a

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 ...

project dedicated to searching 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 ...

s. It is also used in

overclocking In computing, overclocking is the practice of increasing the clock rate of a computer to exceed that certified by the manufacturer. Commonly, operating voltage is also increased to maintain a component's operational stability at accelerated sp ...

to test for system stability. Although most of its source code is available, Prime95 is not

free and open-source software Free and open-source software (FOSS) is software available under a license that grants users the right to use, modify, and distribute the software modified or not to everyone free of charge. FOSS is an inclusive umbrella term encompassing free ...

because its

end-user license agreement 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 ...

states that if the software is used to find a prime qualifying for a bounty 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, ...

, then that bounty will be claimed and distributed by GIMPS.

Finding Mersenne primes by volunteer computing

Prime95 tests numbers for primality using the

Fermat primality test The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Concept Fermat's little theorem states that if ''p'' is prime and ''a'' is not divisible by ''p'', then :a^ \equiv 1 \pmod. If one wants to tes ...

(referred to internally as PRP, or "probable prime"). For much of its history, it used the

Lucas–Lehmer primality test In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 and subsequently proved by Derrick Henry Lehmer in 1930. The test The Lucas–Lehmer test works ...

, but the availability of Lucas–Lehmer assignments was deprecated in April 2021 to increase search throughput. Specifically, to guard against faulty results, every Lucas–Lehmer test had to be performed twice in its entirety, while Fermat tests can be verified in a small fraction of their original run time using a proof generated during the test by Prime95. Current versions of Prime95 remain capable of Lucas–Lehmer testing for the purpose of double-checking existing Lucas–Lehmer results, and for fully verifying "probably prime" Fermat test results (which, unlike "prime" Lucas–Lehmer results, are not conclusive). To reduce the number of full-length primality tests needed, Prime95 first checks numbers for trivial compositeness by attempting to find a small

factor Factor (Latin, ) may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used ...

. As of 2024, test candidates are mainly filtered using

Pollard's p − 1 algorithm Pollard's ''p'' − 1 algorithm is a number theory, number theoretic integer factorization algorithm, invented by John Pollard (mathematician), John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable ...

Trial division Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer , the integer to be factored, can be divided by each number in turn that ...

is implemented, but Prime95 is rarely used for that work in practice because it can be done much more efficiently using a

GPU A graphics processing unit (GPU) is a specialized electronic circuit designed for digital image processing and to accelerate computer graphics, being present either as a discrete video card or embedded on motherboards, mobile phones, personal ...

, due to the type of arithmetic involved. Finally, the elliptic-curve factorization method and

Williams's p + 1 algorithm In computational number theory, Williams's ''p'' + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by Hugh C. Williams in 1982. It works well if the nu ...

are implemented, but are considered not useful at modern GIMPS testing levels and mostly used in attempts to factor much smaller Mersenne numbers that have already undergone primality testing. GIMPS has discovered 18 new Mersenne primes since its foundation in 1996, the first 17 of which using Prime95. The 18th and most recent, M_136279841, was discovered in October 2024 using an

Nvidia Nvidia Corporation ( ) is an American multinational corporation and technology company headquartered in Santa Clara, California, and incorporated in Delaware. Founded in 1993 by Jensen Huang (president and CEO), Chris Malachowsky, and Curti ...

GPU, being the first GIMPS discovery to not have used Prime95 and its

CPU A central processing unit (CPU), also called a central processor, main processor, or just processor, is the primary processor in a given computer. Its electronic circuitry executes instructions of a computer program, such as arithmetic, log ...

computation. 15 of the 17 primes discovered with Prime95 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 the time of their respective discoveries, the exceptions being M_37156667 and M_42643801, which were discovered out of order from the larger M_43112609.

Use for stress testing

To maximize search throughput, most of Prime95 is written in hand-tuned assembly, which makes its system resource usage much greater than most other computer programs. Additionally, due to the high precision requirements of primality testing, the program is very sensitive to computation errors and proactively reports them. These factors make it a commonly used tool among

overclockers In computing, overclocking is the practice of increasing the clock rate of a computer to exceed that certified by the manufacturer. Commonly, operating voltage is also increased to maintain a component's operational stability at accelerated sp ...

to check the stability of a particular configuration.

References

External links

* with downloads for various architectures * How to use Prime95 for stress (torture) testing{{Cite web , title=Torture test your CPU with Prime95 , url=http://www.playtool.com/pages/prime95/prime95.html , access-date=2022-09-15 , website=www.playtool.com Volunteer computing projects Primality tests Benchmarks (computing) Great Internet Mersenne Prime Search Science software for macOS Science software for Windows Science software for Linux BSD software Mathematical software

Finding Mersenne primes by volunteer computing

Use for stress testing

See also

References

External links