Largest known prime number explained

The largest known prime number is, 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 natural number greater than 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, because they can utilize a specialized primality test that is faster than the general one., the six 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 ones, since the binary form of 2^k − 1 is simply k ones.^[5]

Finding larger prime numbers is popularly understood to permit stronger encryption, yet this is incorrect.^{[6] [7]}

Current record

The record is currently held by with 24,862,048 digits, found by GIMPS in December 2018.^[8] The first and last 120 digits of its value are shown below:

, this prime has held the record for more than 5 years, longer than any other prime since M₁₉₉₃₇ (which held the record for 7 years from 1971 to 1978).

Prizes

There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes. A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.^[9] In 2008, a ten-million digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.^[10] Time called this prime the 29th top invention of 2008.^[11]

Both of these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further US$250,000 prize is offered for the first prime with at least one billion digits.

GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.^[12]

History of largest known prime numbers

The following table lists the progression of the largest known prime number in ascending order. Here is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was, which was the largest known prime for 144 years. No records are known prior to 1456.

Number	Decimal expansion (partial for numbers > M1000)	Digits	Year found	Discoverer
M13	8,191	4	1456	Anonymous
M17	131,071	6	1588	Pietro Cataldi
M19	524,287	6	1588	Pietro Cataldi
\tfrac{ 2 ^{32} +1 }{ 641 }	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 2^32 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.^[13]
M31	2,147,483,647	10	1772	Leonhard Euler
\tfrac{ 10 ^{18} +1 }{ 1000001 }	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.
\tfrac{ 2 ^{64} +1 }{ 274177 }	67,280,421,310,721	14	1855	Thomas Clausen (but no proof was provided).	-	[M<sub>59</sub>/179951]	[3,203,431,780,337]	[13]	[1867]	Landry. A record if the immediately preceding entry is excluded.-->
M127	170,141,183,460,469,231,731,687,303,715,884,105,727	39	1876	Édouard Lucas
\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	5210644015679228794060694325390955853335898483908056458352183851018372555735221	79	1951	J. C. P. Miller & D. J. Wheeler^[14] Using Cambridge's EDSAC computer
M521	6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151	157	1952	Raphael M. Robinson
M607	531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127	183	1952	Raphael M. Robinson
M1279	104079321946...703168729087	386	1952	Raphael M. Robinson
M2203	147597991521...686697771007	664	1952	Raphael M. Robinson
M2281	446087557183...418132836351	687	1952	Raphael M. Robinson
M3217	259117086013...362909315071	969	1957	Hans Riesel
M4423	285542542228...902608580607	1,332	1961	Alexander Hurwitz
M9689	478220278805...826225754111	2,917	1963	Donald B. Gillies
M9941	346088282490...883789463551	2,993	1963	Donald B. Gillies
M11213	281411201369...087696392191	3,376	1963	Donald B. Gillies
M19937	431542479738...030968041471	6,002	1971	Bryant Tuckerman
M21701	448679166119...353511882751	6,533	1978	Laura A. Nickel and Landon Curt Noll^[15]
M23209	402874115778...523779264511	6,987	1979	Landon Curt Noll
M44497	854509824303...961011228671	13,395	1979	David Slowinski and Harry L. Nelson
M86243	536927995502...709433438207	25,962	1982	David Slowinski
M132049	512740276269...455730061311	39,751	1983	David Slowinski
M216091	746093103064...103815528447	65,050	1985	David Slowinski
391581 x 2 ^{216193} - 1	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.^[16] ^[17] 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
M859433	129498125604...243500142591	258,716	1994	David Slowinski and Paul Gage
M1257787	412245773621...976089366527	378,632	1996	David Slowinski and Paul Gage
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 the PrimePages,^[18] of which the twenty largest are listed below.^[19]

Rank	Number	Discovered	Digits	Form	Ref
1	2^82589933 − 1	2018-12-07	24,862,048	Mersenne
2	2^77232917 − 1	2017-12-26	23,249,425	Mersenne	^[20]
3	2^74207281 − 1	2016-01-07	22,338,618	Mersenne	^[21]
4	2^57885161 − 1	2013-01-25	17,425,170	Mersenne	^[22]
5	2^43112609 − 1	2008-08-23	12,978,189	Mersenne	^[23]
6	2^42643801 − 1	2009-06-04	12,837,064	Mersenne	^[24]
7	Φ3(−516693^1048576)	2023-10-02	11,981,518	Generalized unique	^[25]
8	Φ3(−465859^1048576)	2023-05-31	11,887,192	Generalized unique	^[26]
9	2^37156667 − 1	2008-09-06	11,185,272	Mersenne
10	2^32582657 − 1	2006-09-04	9,808,358	Mersenne	^[27]
11	10223 × 2^31172165 + 1	2016-10-31	9,383,761	Proth	^[28]
12	2^30402457 − 1	2005-12-15	9,152,052	Mersenne	^[29]
13	2^25964951 − 1	2005-02-18	7,816,230	Mersenne	^[30]
14	2^24036583 − 1	2004-05-15	7,235,733	Mersenne	^[31]
15	1963736^1048576 + 1	2022-09-24	6,598,776	Generalized Fermat	^[32]
16	1951734^1048576 + 1	2022-08-09	6,595,985	Generalized Fermat	^[33]
17	202705 × 2^21320516 + 1	2021-11-25	6,418,121	Proth	^[34]
18	2^20996011 − 1	2003-11-17	6,320,430	Mersenne	^[35]
19	1059094^1048576 + 1	2018-10-31	6,317,602	Generalized Fermat	^[36]
20	3 × 2^20928756 − 1	2023-07-05	6,300,184	Thabit	^[37]

See also

• List of largest known primes and probable primes

External links

• Press release about the largest known prime 2^82,589,933−1
• Press release about the former largest known prime 2^77,232,917−1
• Press release about the former largest known prime 2^74,207,281−1

Notes and References

1. Web site: GIMPS Project Discovers Largest Known Prime Number: 2^82,589,933-1 . 21 December 2018 . Mersenne Research, Inc. . 21 December 2018 .
2. Web site: The largest known primes – Database Search Output . Prime Pages . 19 March 2023.
3. Web site: The Largest Known Prime by Year: A Brief History . Chris . Caldwell . Prime Pages . 19 March 2023.
4. 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 originally by Caldwell.
5. Web site: Perfect Numbers. Penn State University. 6 October 2019. An interesting side note is about the binary representations of those numbers....
6. News: McKinnon . Mika . January 4, 2018 . This Is the Largest Known Prime Number Yet . July 6, 2024 . Smithsonian.
7. Web site: Johnston . Nathaniel . September 11, 2009 . No, Primes with Millions of Digits Are Not Useful for Cryptography . July 6, 2024 . njohnston.ca.
8. Web site: GIMPS Project Discovers Largest Known Prime Number: 2^82,589,933-1 . 21 December 2018 . Mersenne Research, Inc. . 21 December 2018 .
9. Electronic Frontier Foundation, Big Prime Nets Big Prize.
10. Web site: Record 12-Million-Digit Prime Number Nets $100,000 Prize . October 14, 2009 . Electronic Frontier Foundation . . November 26, 2011 .
11. News: Best Inventions of 2008 - 29. The 46th Mersenne Prime . https://web.archive.org/web/20081102044641/http://www.time.com/time/specials/packages/article/0,28804,1852747_1854195_1854157,00.html . dead . November 2, 2008 . Time . . January 17, 2012 . October 29, 2008.
12. Web site: GIMPS by Mersenne Research, Inc. . 21 November 2022 . mersenne.org.
13. Book: How Euler Did Even More. 9780883855843. Edward Sandifer. C.. 19 November 2014. The Mathematical Association of America .
14. Miller . J. C. P. . J. C. P. Miller . 1951 . Large Prime Numbers . Nature . 168 . 4280 . 838 . 1951Natur.168..838M . 10.1038/168838b0.
15. [Landon Curt Noll]
16. Letters to the Editor . 2324686 . The American Mathematical Monthly . 1990 . 97 . 3 . 214–215 . 10.1080/00029890.1990.11995576 . Brown . John . Noll . Landon Curt . Parady . B. K. . Smith . Joel F. . Zarantonello . Sergio E. . Smith . Gene Ward . Robinson . Raphael M. . Andrews . George E. .
17. https://t5k.org/bios/code.php?code=Z Proof-code: Z
18. Web site: The Prime Database: The List of Largest Known Primes Home Page. t5k.org/primes. 19 March 2023.
19. Web site: The Top Twenty: Largest Known Primes. 19 March 2023.
20. Web site: GIMPS Project Discovers Largest Known Prime Number: 2^77,232,917-1. mersenne.org. Great Internet Mersenne Prime Search. 3 January 2018.
21. Web site: GIMPS Project Discovers Largest Known Prime Number: 2^74,207,281-1. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017.
22. Web site: GIMPS Discovers 48th Mersenne Prime, 2^57,885,161-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 5 February 2013.
23. Web site: 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. 29 September 2017. 15 September 2008.
24. Web site: 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. 29 September 2017. 12 April 2009.
25. Web site: PrimePage Primes: Phi(3, - 516693^1048576) . t5k.org.
26. Web site: PrimePage Primes: Phi(3, - 465859^1048576) . t5k.org.
27. Web site: GIMPS Discovers 44th Mersenne Prime, 2^32,582,657-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 11 September 2006.
28. Web site: PrimeGrid's Seventeen or Bust Subproject. primegrid.com. PrimeGrid. 30 September 2017.
29. Web site: GIMPS Discovers 43rd Mersenne Prime, 2^30,402,457-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 24 December 2005.
30. Web site: GIMPS Discovers 42nd Mersenne Prime, 2^25,964,951-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 27 February 2005.
31. Web site: GIMPS Discovers 41st Mersenne Prime, 2^24,036,583-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 28 May 2004.
32. Web site: PrimeGrid's Generalized Fermat Prime Search. primegrid.com. PrimeGrid. 7 October 2022.
33. Web site: PrimeGrid's Generalized Fermat Prime Search. primegrid.com. PrimeGrid. 17 September 2022.
34. Web site: PrimeGrid's Extended Sierpinski Problem Prime Search. primegrid.com. PrimeGrid. 28 December 2021.
35. Web site: GIMPS Discovers 40th Mersenne Prime, 2^20,996,011-1 is now the Largest Known Prime.. mersenne.org. Great Internet Mersenne Prime Search. 29 September 2017. 2 December 2003.
36. Web site: PrimeGrid's Generalized Fermat Prime Search . 7 November 2018 . primegrid.com . PrimeGrid.
37. Web site: PrimeGrid's 321 Prime Search . 17 July 2023 . primegrid.com . PrimeGrid.