Largest known prime number
The largest known prime number (as of December 2020) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.[1]
A prime number is a positive integer, excluding 1, with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two. As of December 2020, 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 very fast compared to other known primality tests for other kinds of numbers. With current computers, a multi-million digit Mersenne-like number can be proven prime, but only multi-thousand digit other numbers can be proven prime. Probable primes, such as repunit R8177207, pass probabilistic primality tests but are not truly proven prime.
Current record[]
The record is currently held by 282,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018.[1] The first and last 120 digits of its value are shown below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...
(24,861,808 digits omitted)
... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]
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 < M1000) |
Digits | Year found | Discoverer (see also Mersenne prime) |
---|---|---|---|---|
M13 | 8,191 | 4 | 1456 | Anonymous |
M17 | 131,071 | 6 | 1588 | Pietro Cataldi |
M19 | 524,287 | 6 | 1588 | Pietro Cataldi |
6,700,417 | 7 | 1732 | Leonhard Euler? Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.[10] | |
M31 | 2,147,483,647 | 10 | 1772 | Leonhard Euler |
999,999,000,001 | 12 | 1851 | Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record. | |
67,280,421,310,721 | 14 | 1855 | Thomas Clausen (but no proof was provided). | |
M127 | 170,141,183,460,469, |
39 | 1876 | Édouard Lucas |
20,988,936,657,440, |
44 | 1951 | with a mechanical calculator; the largest record not set by computer. | |
180×(M127)2+1 |
521064401567922879406069432539 |
79 | 1951 | J. C. P. Miller & D. J. Wheeler[11] Using Cambridge's EDSAC computer |
M521 |
686479766013060971498190079908 |
157 | 1952 | |
M607 |
531137992816767098689588206552 |
183 | 1952 | |
M1279 | 104079321946...703168729087 | 386 | 1952 | |
M2203 | 147597991521...686697771007 | 664 | 1952 | |
M2281 | 446087557183...418132836351 | 687 | 1952 | |
M3217 | 259117086013...362909315071 | 969 | 1957 | |
M4423 | 285542542228...902608580607 | 1,332 | 1961 | |
M9689 | 478220278805...826225754111 | 2,917 | 1963 | |
M9941 | 346088282490...883789463551 | 2,993 | 1963 | |
M11213 | 281411201369...087696392191 | 3,376 | 1963 | |
M19937 | 431542479738...030968041471 | 6,002 | 1971 | Bryant Tuckerman |
M21701 | 448679166119...353511882751 | 6,533 | 1978 | Laura A. Nickel and Landon Curt Noll[12] |
M23209 | 402874115778...523779264511 | 6,987 | 1979 | Landon Curt Noll[12] |
M44497 | 854509824303...961011228671 | 13,395 | 1979 | David Slowinski and Harry L. Nelson[12] |
M86243 | 536927995502...709433438207 | 25,962 | 1982 | David Slowinski[12] |
M132049 | 512740276269...455730061311 | 39,751 | 1983 | David Slowinski[12] |
M216091 | 746093103064...103815528447 | 65,050 | 1985 | David Slowinski[12] |
148140632376...836387377151 | 65,087 | 1989 | A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[13][14] Largest non-Mersenne prime that was the largest known prime when it was discovered. | |
M756839 | 174135906820...328544677887 | 227,832 | 1992 | David Slowinski and Paul Gage[12] |
M859433 | 129498125604...243500142591 | 258,716 | 1994 | David Slowinski and Paul Gage[12] |
M1257787 | 412245773621...976089366527 | 378,632 | 1996 | David Slowinski and Paul Gage[12] |
M1398269 | 814717564412...868451315711 | 420,921 | 1996 | GIMPS, Joel Armengaud |
M2976221 | 623340076248...743729201151 | 895,932 | 1997 | GIMPS, Gordon Spence |
M3021377 | 127411683030...973024694271 | 909,526 | 1998 | GIMPS, Roland Clarkson |
M6972593 | 437075744127...142924193791 | 2,098,960 | 1999 | GIMPS, Nayan Hajratwala |
M13466917 | 924947738006...470256259071 | 4,053,946 | 2001 | GIMPS, Michael Cameron |
M20996011 | 125976895450...762855682047 | 6,320,430 | 2003 | GIMPS, Michael Shafer |
M24036583 | 299410429404...882733969407 | 7,235,733 | 2004 | GIMPS, Josh Findley |
M25964951 | 122164630061...280577077247 | 7,816,230 | 2005 | GIMPS, Martin Nowak |
M30402457 | 315416475618...411652943871 | 9,152,052 | 2005 | GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone |
M32582657 | 124575026015...154053967871 | 9,808,358 | 2006 | GIMPS, Curtis Cooper and Steven Boone |
M43112609 | 316470269330...166697152511 | 12,978,189 | 2008 | GIMPS, Edson Smith |
M57885161 | 581887266232...071724285951 | 17,425,170 | 2013 | GIMPS, Curtis Cooper |
M74207281 | 300376418084...391086436351 | 22,338,618 | 2016 | GIMPS, Curtis Cooper |
M77232917 | 467333183359...069762179071 | 23,249,425 | 2017 | GIMPS, Jonathan Pace |
M82589933 | 148894445742...325217902591 | 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,[15][16] of which the twenty largest are listed below.
Rank | Number | Discovered | Digits | Form | Ref |
---|---|---|---|---|---|
1 | 282589933 − 1 | 2018-12-07 | 24,862,048 | Mersenne | [1] |
2 | 277232917 − 1 | 2017-12-26 | 23,249,425 | Mersenne | [17] |
3 | 274207281 − 1 | 2016-01-07 | 22,338,618 | Mersenne | [18] |
4 | 257885161 − 1 | 2013-01-25 | 17,425,170 | Mersenne | [19] |
5 | 243112609 − 1 | 2008-08-23 | 12,978,189 | Mersenne | [20] |
6 | 242643801 − 1 | 2009-06-04 | 12,837,064 | Mersenne | [21] |
7 | 237156667 − 1 | 2008-09-06 | 11,185,272 | Mersenne | [20] |
8 | 232582657 − 1 | 2006-09-04 | 9,808,358 | Mersenne | [22] |
9 | 10223 × 231172165 + 1 | 2016-10-31 | 9,383,761 | Proth | [23] |
10 | 230402457 − 1 | 2005-12-15 | 9,152,052 | Mersenne | [24] |
11 | 225964951 − 1 | 2005-02-18 | 7,816,230 | Mersenne | [25] |
12 | 224036583 − 1 | 2004-05-15 | 7,235,733 | Mersenne | [26] |
13 | 220996011 − 1 | 2003-11-17 | 6,320,430 | Mersenne | [27] |
14 | 10590941048576 + 1 | 2018-10-31 | 6,317,602 | Generalized Fermat | [28] |
15 | 9194441048576 + 1 | 2017-08-29 | 6,253,210 | Generalized Fermat | [29] |
16 | 168451 × 219375200 + 1 | 2017-09-17 | 5,832,522 | Proth | [30] |
17 | 7 × 218233956 + 1 | 2020-10-01 | 5,488,969 | Proth | [31] |
18 | 1234471048576 − 123447524288 + 1 | 2017-02-23 | 5,338,805 | Generalized unique | [32] |
19 | 7 × 66772401 + 1 | 2019-09-09 | 5,269,954 | [33] | |
20 | 8508301 × 217016603 − 1 | 2018-03-21 | 5,122,515 | Woodall | [34] |
See also[]
- Mersenne prime
- Primality test
- Prime number
- Generalized Fermat prime
- Cullen number
- Woodall number
- Titanic prime
- Gigantic prime
- Megaprime
- Sophie Germain prime
References[]
- ^ Jump up to: a b c "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
- ^ Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved June 3, 2018.
- ^ Jump up to: 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 ⋅ 2216,193 − 1; see also The Largest Known Prime by year: A Brief History by Caldwell.
- ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019.
An interesting side note is about the binary representations of those numbers...
- ^ https://www.mersenne.org/primes/press/M82589933.html
- ^ Jump up to: 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. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
- ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. ISBN 9780883855843.
- ^ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
- ^ Jump up to: a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
- ^ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
- ^ Proof-code: Z, The Prime Pages.
- ^ "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: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
- ^ "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.
- ^ Jump up to: 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.
- ^ "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.
- ^ "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.
- ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
- ^ "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.
- ^ "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.
- ^ "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.
- ^ "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.
- ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
- ^ "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.
- ^ "PrimePage Primes: 7 x 2^18233956 + 1". Retrieved 10 February 2021.
- ^ "The Prime Database: Phi(3,-123447^524288)". primes.utm.edu. The Prime Pages. Retrieved 30 September 2017.
- ^ "The Prime Database: 7*6^6772401+1". primes.utm.edu. The Prime Pages=12 September 2019.
- ^ "PrimeGrid's Woodall Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 2 April 2018.
External links[]
- Prime numbers
- Large integers
- World records
- Superlatives
- Great Internet Mersenne Prime Search