# Largest known prime number

The largest known prime number (as of January 2020) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.[1]

A 2016 plot of the number of digits in largest known prime by year, since the electronic computer. The vertical scale is logarithmic. The red line is the exponential curve of best fit: y = exp(0.187394 t − 360.527), where t is in years.

A prime number is a positive integer with no divisors other than 1 and itself, excluding 1. Euclid recorded a proof 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, numbers that are one less than a power of two. As of December 2018, the eight largest known primes are Mersenne primes.[2] The last seventeen record primes were Mersenne primes.[3][4] The binary representation of any Mersenne prime is composed of all 1's, since the binary form of 2k - 1 is simply k 1's.[5]

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.

## Current record

The record is currently held by 282,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018.[1] Its value is:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...

(24,861,808 digits omitted)

... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]

The first and last 120 digits are shown above.

## Prizes

The Great Internet Mersenne Prime Search (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 for record primes.[7] 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.[8] In 2008 the record passed ten million digits, earning a US\$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.[7] Time called it the 29th top invention of 2008.[9] 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.[7]

## History of largest known prime numbers

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. No records are known before 1456.

Number Decimal expansion
(only for numbers < 1050)
Digits Year found Discoverer
M13 8,191 4 1456 Anonymous
M17 131,071 6 1588 Pietro Cataldi
M19 524,287 6 1588 Pietro Cataldi
${\displaystyle {\tfrac {2^{32}+1}{641}}}$  6,700,417 7 1732 Leonhard Euler
M31 2,147,483,647 10 1772 Leonhard Euler
${\displaystyle {\tfrac {2^{64}+1}{274177}}}$  67,280,421,310,721 14 1855 Thomas Clausen
M127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 Édouard Lucas
${\displaystyle {\tfrac {2^{148}+1}{17}}}$  20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 44 1951 Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1 79 1951 Using Cambridge's EDSAC computer
M521 157 1952
M607 183 1952
M1279 386 1952
M2203 664 1952
M2281 687 1952
M3217 969 1957
M4423 1,332 1961
M9689 2,917 1963
M9941 2,993 1963
M11213 3,376 1963
M19937 6,002 1971
M21701 6,533 1978
M23209 6,987 1979
M44497 13,395 1979
M86243 25,962 1982
M132049 39,751 1983
M216091 65,050 1985
391581×2216193−1 65,087 1989 Largest non-Mersenne prime that was the largest known prime when it was discovered
M756839 227,832 1992
M859433 258,716 1994
M1257787 378,632 1996
M1398269 420,921 1996 GIMPS, Joel Armengaud
M2976221 895,932 1997 GIMPS, Gordon Spence
M3021377 909,526 1998 GIMPS, Roland Clarkson
M6972593 2,098,960 1999 GIMPS, Nayan Hajratwala
M13466917 4,053,946 2001 GIMPS, Michael Cameron
M20996011 6,320,430 2003 GIMPS, Michael Shafer
M24036583 7,235,733 2004 GIMPS, Josh Findley
M25964951 7,816,230 2005 GIMPS, Martin Nowak
M30402457 9,152,052 2005 GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M32582657 9,808,358 2006 GIMPS, Curtis Cooper and Steven Boone
M43112609 12,978,189 2008 GIMPS, Edson Smith
M57885161 17,425,170 2013 GIMPS, Curtis Cooper
M74207281 22,338,618 2016 GIMPS, Curtis Cooper
M77232917 23,249,425 2017 GIMPS, Jonathan Pace
M82589933 24,862,048 2018 GIMPS, Patrick Laroche

GIMPS 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,[10][11] of which the twenty largest are listed below.

Rank Number Discovered Digits Ref
1 282589933 − 1 2018-12-07 24,862,048 [1]
2 277232917 − 1 2017-12-26 23,249,425 [12]
3 274207281 − 1 2016-01-07 22,338,618 [13]
4 257885161 − 1 2013-01-25 17,425,170 [14]
5 243112609 − 1 2008-08-23 12,978,189 [15]
6 242643801 − 1 2009-06-04 12,837,064 [16]
7 237156667 − 1 2008-09-06 11,185,272 [15]
8 232582657 − 1 2006-09-04 9,808,358 [17]
9 10223 × 231172165 + 1 2016-10-31 9,383,761 [18]
10 230402457 − 1 2005-12-15 9,152,052 [19]
11 225964951 − 1 2005-02-18 7,816,230 [20]
12 224036583 − 1 2004-05-15 7,235,733 [21]
13 220996011 − 1 2003-11-17 6,320,430 [22]
14 10590941048576 + 1 2018-10-31 6,317,602 [23]
15 9194441048576 + 1 2017-08-29 6,253,210 [24]
16 168451 × 219375200 + 1 2017-09-17 5,832,522 [25]
17 1234471048576 − 123447524288 + 1 2017-02-23 5,338,805 [26]
18 7 × 66772401 + 1 2019-09-09 5,269,954 [27]
19 8508301 × 217016603 − 1 2018-03-21 5,122,515 [28]
20 99739 × 214019102 + 1 2019-12-24 4,220,176 [29]

## References

1. ^ a b c "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
2. ^ Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved June 3, 2018.
3. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016.
4. ^ The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2216,193 − 1; see also The Largest Known Prime by Year: A Brief History by Caldwell.
5. ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
6. ^ https://www.mersenne.org/primes/press/M82589933.html
7. ^ a b c "Record 12-Million-Digit Prime Number Nets \$100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
8. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
9. ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Retrieved January 17, 2012.
10. ^ "The Prime Database: The List of Largest Known Primes Home Page". primes.utm.edu/primes. Chris K. Caldwell. Retrieved 30 September 2017.
11. ^ "The Top Twenty: Largest Known Primes". Chris K. Caldwell. Retrieved 3 January 2018.
12. ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
13. ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
14. ^ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
15. ^ a b "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
16. ^ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
17. ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
18. ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
19. ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
20. ^ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
21. ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
22. ^ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.
23. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
24. ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
25. ^ "PrimeGrid's Prime Sierpinski Problem" (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.
26. ^ "The Prime Database: Phi(3,-123447^524288)". primes.utm.edu. The Prime Pages. Retrieved 30 September 2017.
27. ^ "The Prime Database: The Prime Database: 7*6^6772401+1". primes.utm.edu. The Prime Pages=12 September 2019.
28. ^ "PrimeGrid's Woodall Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 2 April 2018.
29. ^ "PrimeGrid's Extended Sierpinski Problem Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 16 January 2020.