Largest known prime number
As of January 2016[update], the largest known prime number is 257,885,161 − 1, a number with 17,425,170 digits. It was found in 2013 by the Great Internet Mersenne Prime Search (GIMPS).
Euclid proved that there is no largest prime number, so many mathematicians and hobbyists continue to search for large prime numbers.
Many of the largest known primes are Mersenne primes. As of February 2013[update] the ten largest known primes are Mersenne primes, while the eleventh is the largest known non-Mersenne prime.[1] The last 15 record primes were Mersenne primes.[1]
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
The record is currently held by 257,885,161 − 1 with 17,425,170 digits, discovered by the GIMPS in 2013.[2] The value of it is
- 5818872662322464421751002121132323686363708523254215893257817044
- ... (17,425,042 digits omitted) ...
- 6822494937745410942833323095203705645658725746141988071724285951
The first and last 64 digits are shown above.
Prizes
There are several prizes offered by the Electronic Frontier Foundation for record primes.[3]
The record passed one million digits in 1999, earning a $50,000 prize.[4] In 2008 the record passed ten million digits, earning a $100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.[3] Time called it the 29th top invention of 2008.[5] 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.[3]
History
The following table lists the progression of the largest known prime number in ascending order. 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 | # digits | Year found | Notes | |
|---|---|---|---|---|
| M13 | 8,191 | 4 | 1456 | Anonymous discovery |
| M17 | 131,071 | 6 | 1460 | Anonymous discovery |
| M19 | 524,287 | 6 | 1588 | Found by Pietro Cataldi |
| (232+1)/641 | 6,700,417 | 7 | 1732 | Found by Leonhard Euler |
| M31 | 2,147,483,647 | 10 | 1772 | Found by Leonhard Euler |
| (264+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 |
| (2148+1)/17 | 20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 | 44 | 1951 | Found by Aimé Ferrier; 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 | ||
The ten largest known prime numbers
| Rank | Prime number | Found by | Found date | Number of digits | Reference |
|---|---|---|---|---|---|
| 1st | 257,885,161 − 1 | GIMPS | 2013 January 25 | 17,425,170 | [1] |
| 2nd | 243,112,609 − 1 | GIMPS | 2008 August 23 | 12,978,189 | [1] |
| 3rd | 242,643,801 − 1 | GIMPS | 2009 April 12 | 12,837,064 | [6] |
| 4th | 237,156,667 − 1 | GIMPS | 2008 September 6 | 11,185,272 | [6] |
| 5th | 232,582,657 − 1 | GIMPS | 2006 September 4 | 9,808,358 | [6] |
| 6th | 230,402,457 − 1 | GIMPS | 2005 December 15 | 9,152,052 | [7] |
| 7th | 225,964,951 − 1 | GIMPS | 2005 February 18 | 7,816,230 | [7] |
| 8th | 224,036,583 − 1 | GIMPS | 2004 May 15 | 7,235,733 | [7] |
| 9th | 220,996,011 − 1 | GIMPS | 2003 November 17 | 6,320,430 | [7] |
| 10th | 213,466,917 − 1 | GIMPS | 2001 November 14 | 4,053,946 | [7] |
GIMPS found the eleven latest records on ordinary computers operated by participants around the world.
The largest known prime that is not a Mersenne prime is 19249 × 213018586 + 1, found by Seventeen or Bust.[8] This is currently the eleventh largest known prime.
See also
- Mersenne prime
- Primality test
- Prime number
- Fermat number#Generalized Fermat primes
- Cullen number
- Sophie Germain prime
- Woodall number
References
- ↑ 1.0 1.1 1.2 1.3 Chris Caldwell, The largest known primes. Retrieved on 2013-02-05.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 3.0 3.1 3.2 Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Electronic Frontier Foundation, Big Prime Nets Big Prize.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ 6.0 6.1 6.2 Landon Curt Noll, Mersenne Prime Digits and Names. Retrieved on 2011-01-03.
- ↑ 7.0 7.1 7.2 7.3 7.4 Samuel Yates, Chris Caldwell, The largest known primes. Retrieved on 2014-03-08.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
