As of December 2017, the largest known prime number is 277,232,917 − 1, a number with 23,249,425 digits. It was found on December 26, 2017 by the Great Internet Mersenne Prime Search (GIMPS).

 

Euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.

Many of the largest known primes are Mersenne primes. As of January 2017, the six largest known primes are Mersenne primes. [2] The last 16 record primes were Mersenne primes.[3][3]

The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers.

 

 

The current record

The record is currently held by 277,232,917 − 1 with 23,249,425 digits.[38]

Prizes

The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3000 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 for record primes.[21] GIMPS is also coordinating its long-range search efforts for primes of 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.[13] In 2008 the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.[21] Time called it the 29th top invention of 2008.[8] 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.[21] Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS.

History

The following table lists the progression of the largest known prime number in ascending order.[3] Here Mn= 2n − 1 is the Mersenne number with exponent n. The longest record-holder known was M19 = 524,287, which was the largest known prime for 144 years. Almost no records are known before 1456.

NumberDecimal expansion
(only for numbers < 1050)
DigitsYear foundNotes
(for larger Mersenne primes, see Mersenne prime)
11112~1650 BCEancient Egyptians (disputed)[14]
771~400 BCEIt was known to Philolaus that 7 is a prime[15]
M71273~300 BCEIt was known to Euclid that 127 and 89 are primes[2][17]
M138,19141456Anonymous discovery
M17131,07161460Anonymous discovery
M19524,28761588Found by Pietro Cataldi
6,700,41771732Found by Leonhard Euler
M312,147,483,647101772Found by Leonhard Euler
67,280,421,310,721141855Found by Thomas Clausen
M127170,141,183,460,469,231,731,687,303,715,884,105,727391876Found by Édouard Lucas
20,988,936,657,440,586,486,151,264,256,610,222,593,863,921441951Found by Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1 791951Using Cambridge's EDSAC computer
M521 1571952
M607 1831952
M1279 3861952
M2203 6641952
M2281 6871952
M3217 9691957
M4423 1,3321961
M9689 2,9171963
M9941 2,9931963
M11213 3,3761963
M19937 6,0021971
M21701 6,5331978
M23209 6,9871979
M44497 13,3951979
M86243 25,9621982
M132049 39,7511983
M216091 65,0501985
391581×2216193−1 65,0871989
M756839 227,8321992
M859433 258,7161994
M1257787 378,6321996
M1398269 420,9211996
M2976221 895,9321997
M3021377 909,5261998
M6972593 2,098,9601999
M13466917 4,053,9462001
M20996011 6,320,4302003
M24036583 7,235,7332004
M25964951 7,816,2302005
M30402457 9,152,0522005
M32582657 9,808,3582006
M43112609 12,978,1892008
M57885161 17,425,1702013
M74207281 22,338,6182016

GIMPS found the thirteen 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,[2] the twenty largest are listed below.

RankNumberDiscoveredDigitsRef
1274,207,281 – 12016-01-0722,338,618 
2257,885,161 – 12013-01-2517,425,170[19]
3243,112,609 – 12008-08-2312,978,189[20]
4242,643,801 – 12009-06-0412,837,064[21]
5237,156,667 – 12008-09-0611,185,272[20]
6232,582,657 – 12006-09-049,808,358[22]
710223 ×  231,172,165 + 12016-10-319,383,761[23]
8230,402,457 – 12005-12-159,152,052[24]
9225,964,951 – 12005-02-187,816,230[25]
10224,036,583 – 12004-05-157,235,733[26]
11220,996,011 – 12003-11-176,320,430[27]
12919,4441,048,576 + 12017-08-296,253,210[28]
13168,451 × 219,375,200 + 12017-09-175,832,522[29]
14123,4471,048,576 − 123,447524,288 + 12017-025,338,805[30]
15143,332786,432 − 143,332393,216 + 12017-014,055,114[31]
16213,466,917 − 12001-11-144,053,946[32]
1719249  ×  213,018,586 + 12007-053,918,990[34]
183  ×  211,895,718 − 12015-06-233,580,969[35]
193  ×   211,731,850 − 12015-03-133,531,640[36]
203  ×   211,484,018 − 12014-11-223,457,035[37]

 

See also