# Largest known prime number

The **largest known prime number** (as of March 2018^{[update]}) is 2^{77,232,917} − 1, a number with 23,249,425 digits. It was found by the Great Internet Mersenne Prime Search (GIMPS) in 2017.^{[1]}

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 2018^{[update]}, the seven largest known primes are Mersenne primes.^{[2]} The last 16 record primes were Mersenne primes.^{[3]}^{[4]}

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.

## Contents

## The current record[edit]

The record is currently held by 2^{77,232,917} − 1 with 23,249,425 digits, found by GIMPS in December 2017.^{[1]} Its value is:

467333183359231099988335585561115521251321102817714495798582338593567923480521177207484311099740208849621368090038049317...

(23,249,185 digits omitted)

...285376004518786055402223376672925679282131965467343395945397370476369279894627999939614659217371136582730618069762179071

The first and last 120 digits are shown above.

## Prizes[edit]

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]} 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.^{[5]}

## History[edit]

The following table lists the progression of the largest known prime number in ascending order.^{[3]} Here M_{n}= 2^{n} − 1 is the Mersenne number with exponent *n*. The longest record-holder known was M_{19} = 524,287, which was the largest known prime for 144 years. Almost no records are known before 1456.

Number | Decimal expansion (only for numbers < 10 ^{50}) |
Digits | Year found | Notes (for larger Mersenne primes, see Mersenne prime) |
---|---|---|---|---|

M_{13} |
8,191 | 4 | 1456 | Anonymous discovery |

M_{17} |
131,071 | 6 | 1460 | Anonymous discovery |

M_{19} |
524,287 | 6 | 1588 | Found by Pietro Cataldi |

6,700,417 | 7 | 1732 | Found by Leonhard Euler | |

M_{31} |
2,147,483,647 | 10 | 1772 | Found by Leonhard Euler |

67,280,421,310,721 | 14 | 1855 | Found by Thomas Clausen | |

M_{127} |
170,141,183,460,469,231,731,687,303,715,884,105,727 | 39 | 1876 | Found by Édouard Lucas |

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×(M_{127})^{2}+1 |
79 | 1951 | Using Cambridge's EDSAC computer | |

M_{521} |
157 | 1952 | ||

M_{607} |
183 | 1952 | ||

M_{1279} |
386 | 1952 | ||

M_{2203} |
664 | 1952 | ||

M_{2281} |
687 | 1952 | ||

M_{3217} |
969 | 1957 | ||

M_{4423} |
1,332 | 1961 | ||

M_{9689} |
2,917 | 1963 | ||

M_{9941} |
2,993 | 1963 | ||

M_{11213} |
3,376 | 1963 | ||

M_{19937} |
6,002 | 1971 | ||

M_{21701} |
6,533 | 1978 | ||

M_{23209} |
6,987 | 1979 | ||

M_{44497} |
13,395 | 1979 | ||

M_{86243} |
25,962 | 1982 | ||

M_{132049} |
39,751 | 1983 | ||

M_{216091} |
65,050 | 1985 | ||

391581×2^{216193}−1 |
65,087 | 1989 | ||

M_{756839} |
227,832 | 1992 | ||

M_{859433} |
258,716 | 1994 | ||

M_{1257787} |
378,632 | 1996 | ||

M_{1398269} |
420,921 | 1996 | Found by GIMPS, as were all of those below. | |

M_{2976221} |
895,932 | 1997 | ||

M_{3021377} |
909,526 | 1998 | ||

M_{6972593} |
2,098,960 | 1999 | ||

M_{13466917} |
4,053,946 | 2001 | ||

M_{20996011} |
6,320,430 | 2003 | ||

M_{24036583} |
7,235,733 | 2004 | ||

M_{25964951} |
7,816,230 | 2005 | ||

M_{30402457} |
9,152,052 | 2005 | ||

M_{32582657} |
9,808,358 | 2006 | ||

M_{43112609} |
12,978,189 | 2008 | ||

M_{57885161} |
17,425,170 | 2013 | ||

M_{74207281} |
22,338,618 | 2016 | ||

M_{77232917} |
23,249,425 | 2017 |

GIMPS found the fourteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.

## The twenty largest known prime numbers[edit]

A list of the 5,000 largest known primes is maintained by Chris K. Caldwell,^{[8]}^{[9]} the twenty largest are listed below.

Rank | Number | Discovered | Digits | Ref |
---|---|---|---|---|

1 | 2^{77232917} – 1 |
2017-12-26 | 23,249,425 | ^{[10]} |

2 | 2^{74207281} – 1 |
2016-01-07 | 22,338,618 | ^{[11]} |

3 | 2^{57885161} – 1 |
2013-01-25 | 17,425,170 | ^{[12]} |

4 | 2^{43112609} – 1 |
2008-08-23 | 12,978,189 | ^{[13]} |

5 | 2^{42643801} – 1 |
2009-06-04 | 12,837,064 | ^{[14]} |

6 | 2^{37156667} – 1 |
2008-09-06 | 11,185,272 | ^{[13]} |

7 | 2^{32582657} – 1 |
2006-09-04 | 9,808,358 | ^{[15]} |

8 | 10223 × 2^{31172165} + 1 |
2016-10-31 | 9,383,761 | ^{[16]} |

9 | 2^{30402457} – 1 |
2005-12-15 | 9,152,052 | ^{[17]} |

10 | 2^{25964951} – 1 |
2005-02-18 | 7,816,230 | ^{[18]} |

11 | 2^{24036583} – 1 |
2004-05-15 | 7,235,733 | ^{[19]} |

12 | 2^{20996011} – 1 |
2003-11-17 | 6,320,430 | ^{[20]} |

13 | 919444^{1048576} + 1 |
2017-08-29 | 6,253,210 | ^{[21]} |

14 | 168451 × 2^{19375200} + 1 |
2017-09-17 | 5,832,522 | ^{[22]} |

15 | 123447^{1048576} − 123447^{524288} + 1 |
2017-02 | 5,338,805 | ^{[23]} |

16 | 17016602 × 2^{17016602} − 1 |
2018-03-21 | 5,122,515 | ^{[24]} |

17 | 143332^{786432} − 143332^{393216} + 1 |
2017-01 | 4,055,114 | ^{[25]} |

18 | 2^{13466917} − 1 |
2001-11-14 | 4,053,946 | ^{[26]} |

19 | 19249 × 2^{13018586} + 1 |
2007-05 | 3,918,990 | ^{[27]} |

20 | 3 × 2^{11895718} − 1 |
2015-06-23 | 3,580,969 | ^{[28]} |

## See also[edit]

## References[edit]

- ^
^{a}^{b}"50th Known Mersenne Prime Discovered".*www.mersenne.org*. Retrieved 2018-01-04. **^**Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved January 3, 2018.- ^
^{a}^{b}Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved January 20, 2016. **^**The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2^{216,193}− 1; see also The Largest Known Prime by Year: A Brief History by Caldwell.- ^
^{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. **^**Electronic Frontier Foundation, Big Prime Nets Big Prize.**^**"Best Inventions of 2008 - 29. The 46th Mersenne Prime".*Time*. Time Inc. October 29, 2008. Retrieved January 17, 2012.**^**"The Prime Database: The List of Largest Known Primes Home Page".*primes.utm.edu/primes*. Chris K. Caldwell. Retrieved 30 September 2017.**^**"The Top Twenty: Largest Known Primes". Chris K. Caldwell. Retrieved 3 January 2018.**^**"GIMPS Project Discovers Largest Known Prime Number: 2^{77,232,917}-1".*mersenne.org*. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.**^**"GIMPS Project Discovers Largest Known Prime Number: 2^{74,207,281}-1".*mersenne.org*. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.**^**"GIMPS Discovers 48th Mersenne Prime, 2^{57,885,161}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.- ^
^{a}^{b}"GIMPS Discovers 45th and 46th Mersenne Primes, 2^{43,112,609}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017. **^**"GIMPS Discovers 47th Mersenne Prime, 2^{42,643,801}-1 is newest, but not the largest, known Mersenne Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.**^**"GIMPS Discovers 44th Mersenne Prime, 2^{32,582,657}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.**^**"PrimeGrid's Seventeen or Bust Subproject" (PDF).*primegrid.com*. PrimeGrid. Retrieved 30 September 2017.**^**"GIMPS Discovers 43rd Mersenne Prime, 2^{30,402,457}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.**^**"GIMPS Discovers 42nd Mersenne Prime, 2^{25,964,951}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.**^**"GIMPS Discovers 41st Mersenne Prime, 2^{24,036,583}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.**^**"GIMPS Discovers 40th Mersenne Prime, 2^{20,996,011}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.**^**"PrimeGrid's Generalized Fermat Prime Search" (PDF).*primegrid.com*. PrimeGrid. Retrieved 30 September 2017.**^**"PrimeGrid's Prime Sierpinski Problem" (PDF).*primegrid.com*. PrimeGrid. Retrieved 29 September 2017.**^**"The Prime Database: Phi(3,-123447^524288)".*primes.utm.edu*. The Prime Pages. Retrieved 30 September 2017.**^**"PrimeGrid's Woodall Prime Search" (PDF).*primegrid.com*. PrimeGrid. Retrieved 2 April 2018.**^**"The Prime Database: Phi(3,-143332^393216)".*primes.utm.edu*. The Prime Pages. Retrieved 30 September 2017.**^**"GIMPS Discovers 39th Mersenne Prime, 2^{13,466,917}-1 is now the Largest Known Prime".*mersenne.org*. Great Internet Mersenne Prime Search. 6 December 2001. Retrieved 29 September 2017.**^**"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.**^**"PrimeGrid's 321 Prime Search" (PDF).*primegrid.com*. PrimeGrid. Retrieved 30 September 2017.

## External links[edit]

- Press release about the largest known prime 2
^{77,232,917}−1 - Press release about the former largest known prime 2
^{74,207,281}−1 - Press release about the former largest known prime 2
^{57,885,161}−1 - Press release about the former largest known prime 2
^{43,112,609}−1 - Press release about the former largest known prime 2
^{32,582,657}−1