The largest known prime number () is , a number which has 24,862,048 digits when written in
base 10
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 of the Hindu–Arabic numer ...
. It was found via a computer volunteered by Patrick Laroche 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) in 2018.
![Digits in largest prime found as a function of time](https://upload.wikimedia.org/wikipedia/commons/4/4a/Digits_in_largest_prime_found_as_a_function_of_time.svg)
A
prime number
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 a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, excluding 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 divisible by ...
s other than 1 and itself. According to
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work ''Elements''. There are several proofs of the theorem.
Euclid's proof
Euclid offered ...
there are infinitely many prime numbers, so 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 17t ...
s, numbers that are one less than a power of two, because they can utilise a
specialised primality test that is faster than the general one. , the eight largest known primes are Mersenne primes. The last seventeen record primes were Mersenne primes.
The binary representation of any Mersenne prime is composed of all 1's, since the binary form of 2''
k'' − 1 is simply ''k'' 1's.
Current record
The record is currently held by with 24,862,048 digits, found by
GIMPS
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 ...
in December 2018.
The first and last 120 digits of its value are shown below:
Prizes
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) currently offers a US$3,000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits.
There are several prizes offered by the
Electronic Frontier Foundation
The Electronic Frontier Foundation (EFF) is an international non-profit digital rights group based in San Francisco, California. The foundation was formed on 10 July 1990 by John Gilmore, John Perry Barlow and Mitch Kapor to promote Internet ci ...
for record primes.
GIMPS is also coordinating its long-range search efforts for primes of over 100 million digits and larger and will split the Electronic Frontier Foundation's US$150,000 prize with a winning participant.
The record passed one million digits in 1999, earning a US$50,000 prize. In 2008, the record passed ten million digits, earning a US$100,000 prize and a
Cooperative Computing Award
The Electronic Frontier Foundation (EFF) is an international non-profit digital rights group based in San Francisco, California. The foundation was formed on 10 July 1990 by John Gilmore, John Perry Barlow and Mitch Kapor to promote Internet ci ...
from the
Electronic Frontier Foundation
The Electronic Frontier Foundation (EFF) is an international non-profit digital rights group based in San Francisco, California. The foundation was formed on 10 July 1990 by John Gilmore, John Perry Barlow and Mitch Kapor to promote Internet ci ...
.
''
Time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
'' called this prime the 29th top invention of 2008.
Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS. Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits.
History of largest known prime numbers
![MersennePrimeStamp](https://upload.wikimedia.org/wikipedia/commons/0/05/MersennePrimeStamp.gif)
The following table lists the progression of the largest known prime number in ascending order.
Here is the Mersenne number with exponent ''p''. The longest record-holder known was , which was the largest known prime for 144 years. No records are known prior to 1456.
GIMPS
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 ...
found the fifteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.
The twenty largest known prime numbers
A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, of which the twenty largest are listed below.
See also
*
List of largest known primes and probable primes
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the University of Tennessee's PrimePages and Henri & Renaud Lifchitz' PRP Records. Numbers with more than 2,000,000 digits are shown.
Largest ...
References
External links
Press release about the largest known prime 282,589,933−1Press release about the former largest known prime 277,232,917−1Press release about the former largest known prime 274,207,281−1
{{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 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 ...
Superlatives
Great Internet Mersenne Prime Search
Mersenne primes