# Largest known prime number

As of December 2017, the largest known prime number is 274,207,281 − 1, a number with 22,338,618 digits. It was found in January 2016 by the Great Internet Mersenne Prime Search (GIMPS).

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.

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. [1] The last 16 record primes were Mersenne primes.[2][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 274,207,281 − 1 with 22,338,618 digits, found by GIMPS in 2015 and discovered by humans the following year.[4] Its value is:

300376418084606182052986098359166050056875863030301484843941693345547723219067994296893655300772688320448214882399426727...

(22,338,378 digits omitted)

...717774014762912462113646879425801445107393100212927181629335931494239018213879217671164956287190498687010073391086436351

The first and last 120 digits are shown above.

## 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.[5] 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.[6] In 2008 the record passed ten million digits, earning a US\$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.[5] Time called it the 29th top invention of 2008.[7] 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.[5] 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.[2] 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.

Number Decimal expansion
(only for numbers < 1050)
Digits Year found Notes
(for larger Mersenne primes, see Mersenne prime)
11 11 2 ~1650 BCE ancient Egyptians (disputed)[8]
7 7 1 ~400 BCE It was known to Philolaus that 7 is a prime[9]
M7 127 3 ~300 BCE It was known to Euclid that 127 and 89 are primes[10][11]
M13 8,191 4 1456 Anonymous discovery
M17 131,071 6 1460 Anonymous discovery
M19 524,287 6 1588 Found by Pietro Cataldi
${\displaystyle {\tfrac {2^{32}+1}{641}}}$ 6,700,417 7 1732 Found by Leonhard Euler
M31 2,147,483,647 10 1772 Found by Leonhard Euler
${\displaystyle {\tfrac {2^{64}+1}{274177}}}$ 67,280,421,310,721 14 1855 Found by Thomas Clausen
M127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 Found by 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 Found by 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
M756839 227,832 1992
M859433 258,716 1994
M1257787 378,632 1996
M1398269 420,921 1996
M2976221 895,932 1997
M3021377 909,526 1998
M6972593 2,098,960 1999
M13466917 4,053,946 2001
M20996011 6,320,430 2003
M24036583 7,235,733 2004
M25964951 7,816,230 2005
M30402457 9,152,052 2005
M32582657 9,808,358 2006
M43112609 12,978,189 2008
M57885161 17,425,170 2013
M74207281 22,338,618 2016

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,[12] the twenty largest are listed below.

Rank Number Discovered Digits Ref
1 274,207,281 – 1 2016-01-07 22,338,618 [13]
2 257,885,161 – 1 2013-01-25 17,425,170 [14]
3 243,112,609 – 1 2008-08-23 12,978,189 [15]
4 242,643,801 – 1 2009-06-04 12,837,064 [16]
5 237,156,667 – 1 2008-09-06 11,185,272 [15]
6 232,582,657 – 1 2006-09-04 9,808,358 [17]
7 10223 ×  231,172,165 + 1 2016-10-31 9,383,761 [18]
8 230,402,457 – 1 2005-12-15 9,152,052 [19]
9 225,964,951 – 1 2005-02-18 7,816,230 [20]
10 224,036,583 – 1 2004-05-15 7,235,733 [21]
11 220,996,011 – 1 2003-11-17 6,320,430 [22]
12 919,4441,048,576 + 1 2017-08-29 6,253,210 [23]
13 168,451 × 219,375,200 + 1 2017-09-17 5,832,522 [24]
14 123,4471,048,576 − 123,447524,288 + 1 2017-02 5,338,805 [25]
15 143,332786,432 − 143,332393,216 + 1 2017-01 4,055,114 [26]
16 213,466,917 − 1 2001-11-14 4,053,946 [27]
17 19249  ×  213,018,586 + 1 2007-05 3,918,990 [28]
18 3  ×  211,895,718 − 1 2015-06-23 3,580,969 [29]
19 3  ×   211,731,850 − 1 2015-03-13 3,531,640 [30]
20 3  ×   211,484,018 − 1 2014-11-22 3,457,035 [31]

## References

1. ^ Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved January 20, 2016.
2. ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016.
3. ^ 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.
4. ^ "Mersenne Prime Number discovery - 274207281-1". Great Internet Mersenne Prime Search.
5. ^ 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.
6. ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
7. ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Retrieved January 17, 2012.
8. ^ There is no mentioning among the ancient Egyptians of prime numbers, and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. "The Egyptians used (\$) in the table above for the first primes r = 3, 5, 7, or 11 (also for r = 23). Here is another intriguing observation: That the Egyptians stopped the use of (\$) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/n Table [Retrieved 2012-11-11].
9. ^ Harris, H. S. The Reign of the Whirlwind, 1999 (p.252)
10. ^ Nicomachus' "Introduction to Arithmetic" translated by Martin Luther D'Ooge (p.52)
11. ^ "Euclid's Elements, Book IX, Proposition 36".
12. ^ "The Prime Database: The List of Largest Known Primes Home Page". primes.utm.edu/primes. Chris K. Caldwell. Retrieved 30 September 2017.
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 30 September 2017.
24. ^ "PrimeGrid's Prime Sierpinski Problem" (PDF). primegrid.com. PrimeGrid. Retrieved 29 September 2017.
25. ^ "The Prime Database: Phi(3,-123447^524288)". primes.utm.edu. The Prime Pages. Retrieved 30 September 2017.
26. ^ "The Prime Database: Phi(3,-143332^393216)". primes.utm.edu. The Prime Pages. Retrieved 30 September 2017.
27. ^ "GIMPS Discovers 39th Mersenne Prime, 213,466,917-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 6 December 2001. Retrieved 29 September 2017.
28. ^ "Press release dated May 5, 2007". seventeenorbust.com. Seventeen or Bust project. 5 May 2007. Archived from the original on 30 June 2007. Retrieved 30 September 2017.
29. ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
30. ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
31. ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
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