The largest known top number (as of January 2022) is 2^82,589,933 − 1, a number which has 24,862,048 digits when written in immoral 10. It used to be chanced on by technique of a pc volunteered by Patrick Laroche of the Wide Net Mersenne High Search (GIMPS) in 2018.^[1]

A top number is a obvious integer, excluding 1, without a divisors diversified than 1 and itself. Basically based on Euclid’s theorem there are infinitely many top numbers, so there might be not any such thing as a largest top.

Many of the largest known primes are Mersenne primes, numbers that are one no longer as much as a vitality of two, on fable of they’ll utilise a specialised primality take a look at that’s quicker than the classic one. As of December 2020, the eight largest known primes are Mersenne primes.^[2] The final seventeen memoir primes were Mersenne primes.^[3][4] The binary representation of any Mersenne top is mild of all 1’s, since the binary get of two^k − 1 is purely k 1’s.^[5]

The quick Fourier transform implementation of the Lucas–Lehmer primality take a look at for Mersenne numbers is terribly snappy when in comparison with diversified known primality tests for diversified forms of numbers. With most unique pc programs, a multi-million digit Mersenne-delight in number would be confirmed top, however handiest multi-thousand digit diversified numbers would be confirmed top. Attainable primes, such because the immoral-10 repunit R_8177207, cross probabilistic primality tests however are no longer indubitably confirmed top.

Most unique memoir[edit]

The memoir is for the time being held by 2^82,589,933 − 1 with 24,862,048 digits, chanced on by GIMPS in December 2018.^[1] The first and final 120 digits of its sign are confirmed below:

148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 …

(24,861,808 digits pushed aside)

… 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591^[6]

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Prizes[edit]

The Wide Net Mersenne High Search (GIMPS) for the time being offers a US$3,000 study discovery award for members who gather and trudge their free instrument and whose pc discovers a brand novel Mersenne top having fewer than 100 million digits.

There are several prizes equipped by the Electronic Frontier Foundation for memoir primes.^[7] GIMPS is additionally coordinating its prolonged-fluctuate search efforts for primes of 100 million digits and bigger and could additionally fair split the Electronic Frontier Foundation’s US$150,000 prize with a winning participant.

The memoir passed one million digits in 1999, incomes a US$50,000 prize.^[8] In 2008, the memoir passed ten million digits, incomes a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.^[7] Time known as it the 29th top invention of 2008.^[9] Each the US$50,000 and the US$100,000 prizes were received by participation in GIMPS. Extra prizes are being equipped for the principle top number chanced on with on the least one hundred million digits and the principle with on the least one billion digits.^[7]

Historical previous of largest known top numbers[edit]

The next table lists the progression of the largest known top number in ascending reveal.^[3] Here M_p = 2^p − 1 is the Mersenne number with exponent p. The longest memoir-holder known used to be M₁₉ = 524,287, which used to be the largest known top for 144 years. No records are known sooner than 1456.

Number	Decimal expansion (handiest for numbers < M1000)	Digits	one year chanced on	Discoverer
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 no longer explicitly post the primality of 6,700,417, however the ways he had aged to factorise 232 + 1 intended that he had already accomplished numerous the work most essential to level this, and some experts middle of attention on he knew of it.[10]
M31	2,147,483,647	10	1772	Leonhard Euler
	999,999,000,001	12	1851	Included (however demand-marked) in a checklist of primes by Looff. Given his uncertainty, some attain no longer consist of this as a memoir.
	67,280,421,310,721	14	1855	Thomas Clausen (however no proof used to be equipped).
M127	170,141,183,460,469,231,731,687,303,715,884,105,727	39	1876	Édouard Lucas
	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 memoir no longer build by pc.
180×(M127)2+1	5210644015679228794060694325390955853335898483908056458352183851018372555735221	79	1951	J. C. P. Miller & D. J. Wheeler[11] Utilizing Cambridge’s EDSAC pc
M521	6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151	157	1952
M607	531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127	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 crew, “Amdahl Six”: John Brown, Landon Curt Noll, B. Okay. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[13][14] Ideal non-Mersenne top that used to be the largest known top when it used to be chanced on.
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, College 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 Trek
M82589933	148894445742…325217902591	24,862,048	2018	GIMPS, Patrick Laroche

GIMPS chanced on the fifteen most unique records (all of them Mersenne primes) on extraordinary pc programs operated by members at some level of the field.

The twenty largest known top numbers[edit]

A checklist of the 5,000 largest known primes is maintained by Chris Okay. Caldwell,^[15][16] of which the twenty largest are listed below.

Obnoxious	Number	Found out	Digits	Comprise	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	202705 × 221320516 + 1	2021-12-01	6,418,121	Proth	[27]
14	220996011 − 1	2003-11-17	6,320,430	Mersenne	[28]
15	10590941048576 + 1	2018-10-31	6,317,602	Generalized Fermat	[29]
16	9194441048576 + 1	2017-08-29	6,253,210	Generalized Fermat	[30]
17	168451 × 219375200 + 1	2017-09-17	5,832,522	Proth	[31]
18	69 × 218831865 − 1	2021-12-16	5,668,959
19	7 × 218233956 + 1	2020-10-01	5,488,969	Proth	[32]
20	3  ×  218196595 − 1	2022-01-18	5,477,722	321

Notice additionally[edit]

References[edit]

External hyperlinks[edit]

• Press originate about the largest known top 2^82,589,933−1
• Press originate about the broken-down largest known top 2^77,232,917−1
• Press originate about the broken-down largest known top 2^74,207,281−1