The largest known prime number is , a number which has 41,024,320 digits when written in the

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

system. It was found on October 12, 2024, on a cloud-based

virtual machine In computing, a virtual machine (VM) is the virtualization or emulator, emulation of a computer system. Virtual machines are based on computer architectures and provide the functionality of a physical computer. Their implementations may involve ...

volunteered by Luke Durant, a 36-year-old researcher from San Jose, California, to 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).

Digits in largest prime found as a function of time

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

is a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

greater than 1 with no

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s other than 1 and itself.

Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proven by Euclid in his work ''Euclid's Elements, Elements''. There are several proof ...

proves that for any given prime number, there will always be a higher one, and thus there are infinitely many; there is no largest prime. Many of the largest known primes are

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, numbers that are one less than a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^ ...

, because they can utilize a specialized primality test that is faster than the general one. , the seven largest known primes are Mersenne primes. The last eighteen record primes were Mersenne primes. The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2''^k'' − 1 is simply ''k'' ones. Finding larger prime numbers is sometimes presented as a means to stronger

encryption In Cryptography law, cryptography, encryption (more specifically, Code, encoding) is the process of transforming information in a way that, ideally, only authorized parties can decode. This process converts the original representation of the inf ...

, but this is not the case.

Current record

The record is currently held by with 41,024,320 digits, found by GIMPS on October 12, 2024. The first and last 100 digits of its value are:

Prizes

There are several prizes 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, ...

(EFF) for record primes. A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize. In 2008, a ten-million-digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF. ''

Time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...

'' called this prime the 29th top invention of 2008. Both of these primes were discovered through the

(GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further US$250,000 prize is offered for the first prime with at least one billion digits. GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.

History

The following table lists the progression of the largest known prime number in ascending order. Here is the Mersenne number with exponent ''p'', where ''p'' is a prime number. The longest record-holder known was , which was the largest known prime for 144 years. The primes up to and including

\tfrac

are found without a computer, while the primes starting with 180×(M₁₂₇)²+1 are found using computers. GIMPS volunteers found the sixteen latest records, all of them

s. They were found on ordinary

personal computer A personal computer, commonly referred to as PC or computer, is a computer designed for individual use. It is typically used for tasks such as Word processor, word processing, web browser, internet browsing, email, multimedia playback, and PC ...

s until the most recent one, found by ex-

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 ...

employee Luke Durant using a network of thousands of dedicated

graphics processing unit 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 ...

s (GPUs). Durant spent about one year and US$2 million on the hunt. This is the first time a Mersenne prime has been discovered using GPUs instead of

central processing units 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 ...

(CPUs).

Twenty largest

A list of the 5,000 largest known primes is maintained by the PrimePages, of which the twenty largest are listed below.

References

External links

Press release about the former largest known prime 2^74,207,281−1Press release about the former largest known prime 2^77,232,917−1Press release about the former largest known prime 2^82,589,933−1Press release about the largest known prime 2^136,279,841−1
{{Large numbers Prime numbers Large integers

Prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...