Largest known prime number

As of January 2016, the largest known prime number is 2^74,207,281 − 1, a number with 22,338,618 decimal digits. It was found in 2016 by the Great Internet Mersenne Prime Search (GIMPS).

Plot of the number of digits in largest known prime by year, since the electronic computer. Note that the vertical scale is logarithmic. The red line is the exponential curve of best fit: y = exp(0.187394 t - 360.527), where t is in years.

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 November 2016, the six largest known primes are Mersenne primes, while the seventh is the largest known non-Mersenne prime.^[1] The last 16 record primes were Mersenne primes.^[2]^[3]

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.

The current record

The record is currently held by 2^74,207,281 − 1 with 22,338,618 digits, discovered by the GIMPS in 2016.^[4] Its value is:

300376418084606182052986098359166050056875863030301484843941693345547723219067994296893655300772688320448214882399426831
... (22,338,378 digits omitted) ...

717774014762912462113646879425801445107393100212927181629335931494239018213879217671164956287190498687010073391086436351

The first and last 120 digits are shown above.

Prizes

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 $50,000 prize.^[6] In 2008 the record passed ten million digits, earning a $100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.^[5] Time called it the 29th top invention of 2008.^[7] 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] Both the $50,000 and the $100,000 prizes were won by participation in GIMPS.

History

The following table lists the progression of the largest known prime number in ascending order.^[2] Here M_n= 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. Almost no records are known before 1456.

Number	Decimal expansion (only for numbers < 10⁵⁰)	Digits	Year found	Notes (for larger Mersenne primes, see Mersenne prime)
11	11	2	~1650 BCE	ancient Egyptians (disputed)^[8]
7	7	1	~400 BCE	It was known to Philolaus that 7 is a prime^[9]
127	127	3	~300 BCE	It was known to Euclid that 127 and 89 are primes^[10]^[11]
M₁₃	8,191	4	1456	Anonymous discovery
M₁₇	131,071	6	1460	Anonymous discovery
M₁₉	524,287	6	1588	Found by Pietro Cataldi
${\tfrac {2^{32}+1}{641}}$	6,700,417	7	1732	Found by Leonhard Euler
M₃₁	2,147,483,647	10	1772	Found by Leonhard Euler
${\tfrac {2^{64}+1}{274177}}$	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
${\tfrac {2^{148}+1}{17}}$	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
M_1257787		378,632	1996
M_1398269		420,921	1996
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

The twenty largest known prime numbers

Rank	Prime number	Discovery date	Digits
1	2^74207281 – 1	2016-01-07	22,338,618
2	2^57885161 – 1	2013-01-25	17,425,170
3	2^43112609 – 1	2008-08-23	12,978,189
4	2^42643801 – 1	2009-04-12	12,837,064
5	2^37156667 – 1	2008-09-06	11,185,272
6	2^32582657 – 1	2006-09-04	9,808,358
7	10223×2^31172165 + 1	2016-11-06	9,383,761
8	2^30402457 – 1	2005-12-15	9,152,052
9	2^25964951 – 1	2005-02-18	7,816,230
10	2^24036583 – 1	2004-05-15	7,235,733
11	2^20996011 – 1	2003-11-17	6,320,430
12	2^13466917 – 1	2001-11-14	4,053,946
13	19249×2^13018586 + 1	2007-03-26	3,918,990
14	3×2^11895718 − 1	2015-06-23^[12]	3,580,969
15	3×2^11731850 − 1	2015-03-13	3,531,640
16	3×2^11484018 − 1	2014-11-22	3,457,035
17	3×2^10829346 + 1	2014-01-14	3,259,959
18	475856⁵²⁴²⁸⁸ + 1	2012-08-08	2,976,633
19	356926⁵²⁴²⁸⁸ + 1	2012-06-20	2,911,151
20	341112⁵²⁴²⁸⁸ + 1	2012-06-20	2,900,832

GIMPS found the thirteen latest records on ordinary computers operated by participants around the world

References

↑ Caldwell, Chris. "The largest known primes - Database Search Output". Prime Pages. Retrieved January 20, 2016.
1 2 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.
↑ "Mersenne Prime Number discovery - 2^74207281-1". Great Internet Mersenne Prime Search.
1 2 3 "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
↑ Electronic Frontier Foundation, Big Prime Nets Big Prize.
↑ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Retrieved January 17, 2012.
↑ There is no mentioning among the ancient Egyptians of prime numbers, and they did not have any concept for prime numbers known today. In the Rhind papyrus (1650 BC) the Egyptian fraction expansions have fairly different forms for primes and composites, so it may be argued that they knew about prime numbers. "The Egyptians used ($) in the table above for the first primes $r$ = 3, 5, 7, or 11 (also for $r$ = 23). Here is another intriguing observation: That the Egyptians stopped the use of ($) at 11 suggests they understood (at least some parts of) Eratosthenes's Sieve 2000 years before Eratosthenes 'discovered' it." The Rhind 2/ $n$ Table [Retrieved 2012-11-11].
↑ Harris, H. S. The Reign of the Whirlwind, 1999 (p.252)
↑ Nicomachus' "Introduction to Arithmetic" translated by Martin Luther D'Ooge (p.52)
↑ "Euclid's Elements, Book IX, Proposition 36".
↑ "PrimeGrid's 321 Prime Search" (PDF). Retrieved October 12, 2016.

External links

Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap

First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229

List of prime numbers

Large numbers

Examples in numerical order

Expression methods

Notations	Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation

Operators	Hyperoperation Tetration Pentation Ackermann function Bowers' operators

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