Largest known prime number

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As of January 2014, the largest known prime number is 2^57,885,161 − 1,^[1] a number with 17,425,170 digits.

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(0.188439 t - 362.591), where t is in years.

Euclid proved that there is no largest prime number. However, many mathematicians and hobbyists search for large prime numbers.

Many of the largest known primes are Mersenne primes. As of February 2013 the ten largest known primes are Mersenne primes, while the eleventh is the largest known non-Mersenne prime.^[2] The last 15 record primes were Mersenne primes.^[2]

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 2^57,885,161 − 1 with 17,425,170 digits. Its discovery resulted from the Great Internet Mersenne Prime Search (GIMPS).

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 M_n= 2ⁿ − 1 is the Mersenne number with exponent n.

Number	Digits	Year found
M₁₂₇	39	1876
180×(M₁₂₇)² + 1	79	1951
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

The ten largest known prime numbers

Rank	Prime number	Found by	Found date	Number of digits	Reference
1st	2^57,885,161 − 1	GIMPS	2013 January 25	17,425,170	^[2]
2nd	2^43,112,609 − 1	GIMPS	2008 August 23	12,978,189	^[2]
3rd	2^42,643,801 − 1	GIMPS	2009 April 12	12,837,064	^[6]
4th	2^37,156,667 − 1	GIMPS	2008 September 6	11,185,272	^[6]
5th	2^32,582,657 − 1	GIMPS	2006 September 4	9,808,358	^[6]
6th	2^30,402,457 − 1	GIMPS	2005 December 15	9,152,052	^[7]
7th	2^25,964,951 − 1	GIMPS	2005 February 18	7,816,230	^[7]
8th	2^24,036,583 − 1	GIMPS	2004 May 15	7,235,733	^[7]
9th	2^20,996,011 − 1	GIMPS	2003 November 17	6,320,430	^[7]
10th	2^13,466,917 − 1	GIMPS	2001 November 14	4,053,946	^[7]

GIMPS found the 11 latest records on ordinary computers operated by participants around the world.

References

↑ "GIMPS Project Discovers Largest Known Prime Number, 2^57,885,161-1". Mersenne Research, Inc.
↑ 2.0 2.1 2.2 2.3 Chris Caldwell, The largest known primes. Retrieved on 2013-02-05.
↑ 3.0 3.1 3.2 "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. Retrieved January 17, 2012.
↑ 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 2012-06-09.

External links

v t e 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) 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 Motzkin Bell Partitions Pell Perrin Newman–Shanks–Williams

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

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

Patterns	Twin (p, p + 2) Bi-twin chain (p − 1, p + 1, 2p − 1, 2p + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) 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 Formula for primes Prime gap

List of prime numbers

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