All researchers of Fermat numbers, who found at least one factor
| N | Complete name and links to the biography or personal sites. |
Years of life | Found divisors | Year of the first mention |
Country | Portrait |
|---|---|---|---|---|---|---|
Manual calculations |
||||||
| 1 | Pierre Fermat | (1601-1665) | 5 prime | 1640 | France |  |
| 2 | Leonhard Euler | (1707-1783) | 2 | 1732 | Switzerland/Russia |  |
| 3 | Thomas Clausen | (1801-1885) | 2 | 1855 | Denmark/Russia |  |
| 4 | Edouard Lucas | (1842-1891) | 1 | 1877 | France |  |
| 5 | Ivan Mikheevich Pervushin | (1829-1900) | 2 | 1877 | Russia |  |
| 6 | Fortuné Landry | (1798-1895) | 2 | 1880 | France | |
| 7 | Paul Peter Heinrich Seelhoff | (1829-1896) | 1 | 1886 | Germany | |
| 8 | Allan J.C. Cunningham | (1842-1928) | 3 | 1899 | England | |
| 9 | Alfred Edward Western | (1873-1961) | 5 | 1903 | England | |
| 10 | James Cullen | (1867-1933) | 1 | 1903 | Ireland | |
| 11 | J.C.Morehead | (1877-1950) | 1 | 1906 | ||
| 12 | Maurice Borisovich Kraitchik | (1882-1957) | 1 | 1925 | Belgium |  |
|
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| 13 | John L. Selfridge | (1927-2010) | 6 | 1953 | USA |  |
| 14 | Raphael Mitchel Robinson | (1911-1994) | 20 | 1956 | USA |  |
| 15 | John Brillhart | (1930 - 2022) | 5 | 1962 | USA |  |
| 16 | Hans Riesel | (1929-2014 ) | 1 | 1962 | Sweden |  |
| 17 | Claude P. Wrathall | 11 | 1963 | USA | ||
| 18 | Michael A.Morrison | 2 | 1970 | |||
| 19 | John C. Hallyburton | 2 | 1974 | |||
| 20 | Greg Matthew | 2 | 1976 | |||
| 21 | Hugh Cowie Williams | (1943 - ) | 6 | 1976 | Canada |  |
| 22 | David E.Shippee | (1946-2003) | 4 | 1977 | USA | |
| 23 | Gary B.Gostin | (1954 - ) | 91 | 1978 | USA | |
| 24 | Wilfrid Keller | (1937 - ) | 13 | 1978 | Argentina/Germany | |
| 25 | Robert J. Baillie | 5 | 1979 | |||
| 26 | Philip B. McLaughlin | 7 | 1979 | USA | ||
| 27 | Hiromi Suyama | 7 | 1979 | Japan | ||
| 28 | A.Oliver L.Atkin | (1925-2008) | 5 | 1979 | England |  |
| 29 | N.W.Rickert | 5 | 1979 | |||
| 30 | Gordon V. Cormack | (1956 - ) | 4 | 1979 | Canada |  |
| 31 | Richard Peirce Brent | (1946 - ) | 8 | 1980 | Australia |  |
| 32 | John Michael Pollard | (1941 - ) | 2 | 1980 | England | |
| 33 | Francois Morain | 1 | 1988 | France |  |
|
| 34 | Arjen Klaas Lenstra | (1956 - ) | 2 | 1990 | Holland |  |
| 35 | Mark S.Manasse | 2 | 1990 | USA |  |
|
| 36 | Richard E. Crandall | (1947 - 2012) | 5 | 1991 | USA |  |
| 37 | Harvey Dubner | (1928 - 2019) | 8 | 1992 | USA |  |
| 38 | Jeffrey Young | 8 | 1993 | USA | ||
| 39 | Tadashi Taura | (1954 -) | 18 | 1995 | Japan |  |
| 40 | Karl Dilcher | (1954 - ) | 1 | 1996 | Germany/Canada |  |
| 41 | Chris Van Halewyn | 1 | 1997 | |||
| 42 | Patrick Demichel | 1 | 1997 | France |  |
|
| 43 | Yves Gallot | (1966 - ) | 28 | 1997 | France | |
| 44 | Robert Prethaler | 1 | 1998 |  |
||
| 45 | Gennady Nikolaevich Gusev | (1954 - ) | 1 | 1998 | Russia |  |
| 46 | Richard McIntosh | 1 | 1999 | |||
| 47 | Claude Tardif | 1 | 1999 |  |
||
| 48 | Charles F. Kerchner III | 3 | 1999 |  |
||
| 49 | Dan Morenus | 1 | 1999 |  |
||
| 50 | John Renze | 1 | 1999 | |||
| 51 | John Berchmans Cosgrave | (1946 - ) | 3 | 1999 | Ireland |  |
| 52 | Rachel Lewis | 1 | 2000 | USA | ||
| 53 | Leonid Nikolaevich Durman | (1975 - ) | 23 | 2000 | Russia |  |
| 54 | James Ray Ballinger | (1953 - 2014) | 1 | 2000 | USA |  |
| 55 | Nestor Sergio de Araujo Melo | (1971 - ) | 2 | 2000 | Brazil |  |
| 56 | Payam Samidoost | (1969 - ) | 5 | 2000 | Iran |  |
| 57 | Takahiro Nohara | (1969 - ) | 13 | 2001 | Japan |  |
| 58 | Peter Grobstich | (1942 - ) | 2 | 2001 | Germany |  |
| 59 | Alexander Kruppa | (1974 - ) | 2 | 2001 | Germany |  |
| 60 | Tony Forbes | (1944 - ) | 1 | 2001 | England | |
| 61 | Marko Bodschwinna | (1974 - ) | 1 | 2001 | Germany | |
| 62 | Göran Axelsson | (1966 - ) | 2 | 2001 | Sweden |  |
| 63 | Paul Jobling | (1964 - ) | 20 | 2001 | England |  |
| 64 | George Woltman | (1957 - ) | 26 | 2001 | USA |  |
| 65 | Jim Fougeron | (1966 - ) | 19 | 2001 | USA | |
| 66 | Vasily Anatol'evich Danilov | (1957 - ) | 5 | 2001 | Russia |  |
| 67 | Anton Petrovich Oleynik | (1972 - ) | 1 | 2001 | Russia |  |
| 68 | Dmitry Komin | (1971 - ) | 1 | 2002 | Russia | |
| 69 | Kevin Odermatt | 1 | 2002 | |||
| 70 | Sergey Kuzmin | 1 | 2003 | Russia | ||
| 71 | Craig Kitchen | 1 | 2003 | USA | ||
| 72 | Asko Vuori | (1965 - ) | 2 | 2005 | Finland | |
| 73 | Maximilian Pacher | (1971 - ) | 6 | 2005 | Austria | |
| 74 | Michael Eaton | 2 | 2005 | USA | ||
| 75 | Jun Tajima | 1 | 2005 | Japan | ||
| 76 | Curtis Cooper | 2 | 2005 | USA | ||
| 77 | Reto Keiser | 1 | 2007 | Switzerland | ||
| 78 | Jean Penné | 10 | 2007 | France | ||
| 79 | Pavlos Saridis | (1980 - ) | 1 | 2007 | Greece | |
| 80 | Souichi Murata | 1 | 2007 | Japan | ||
| 81 | Martin Ptáček | (1985 - ) | 1 | 2008 | Czech Republic | |
| 82 | Eric Ueda | 1 | 2008 | USA | ||
| 83 | Senji Yamashita | 1 | 2009 | Japan | ||
| 84 | Eric Embling | 1 | 2009 | USA | ||
| 85 | David Bessell, GIMPS et al. | 3 | 2009 | Australia | ||
| 86 | Sergei Maiorov | (1969 - ) | 1 | 2010 | Russia | |
| 87 | Tapio Rajala | (1982 - ) | 3 | 2010 | Finland | |
| 88 | Cedric Vonck | (1980 - ) | 1 | 2010 | Belgium |  |
| 89 | Michael Vang | 1 | 2010 | USA | ||
| 90 | Peter Strasser | 4 | 2010 | Austria |  |
|
| 91 | Alexey Komkov | 1 | 2010 | Russia | ||
| 92 | Dirk D'huyvetters & PrimeGrid | 1 | 2011 | Belgium | ||
| 93 | Roman Maznichenko | 8 | 2011 | Russia | ||
| 94 | Scott Brown & PrimeGrid | 1 | 2011 | USA | ||
| 95 | Mark Doom & PrimeGrid | 1 | 2011 | USA | ||
| 96 | Nikolay Kamenyuk | 1 | 2011 | Ukraine | ||
| 97 | Peter Doggart & PrimeGrid | 1 | 2011 | England | ||
| 98 | Grzegorz Granowski & PrimeGrid | 1 | 2011 | Poland | ||
| 99 | Bruce Dodson & PrimeGrid | 1 | 2012 | USA | ||
| 100 | Rob Derrera & PrimeGrid | 1 | 2012 | USA | ||
| 101 | Andriy Sen | 1 | 2012 | Ukraine | ||
| 102 | James Boerner & PrimeGrid | 1 | 2012 | USA | ||
| 103 | Robert Boniecki & PrimeGrid | 1 | 2012 | USA | ||
| 104 | Michael Dangler | 1 | 2012 | USA |  |
|
| 105 | Serge Batalov | 9 | 2012 | USA | ||
| 106 | Ryan Propper | 4 | 2013 | USA | ||
| 107 | Marshall Bishop & PrimeGrid | 1 | 2013 | USA | ||
| 108 | Bart van Rooijen & PrimeGrid | 1 | 2013 | Holland | ||
| 109 | Say Yik Tang & PrimeGrid | 1 | 2013 | Malaysia | ||
| 110 | Raymond Ottusch & PrimeGrid | 1 | 2014 | USA | ||
| 111 | Jay Parangalan & PrimeGrid | 1 | 2015 | USA | ||
| 112 | Adolf Nordin | 1 | 2016 | South Africa | ||
| 113 | Mark Rodenkirch | 1 | 2018 | USA | ||
| 114 | James Scott Brown and PrimeGrid | 2 | 2020 | USA | ||
| 115 | Tom Greer and PrimeGrid | 1 | 2021 | USA | ||
| 116 | Luigi Morelli | (1962 - ) | 1 | 2021 | Italy |  |
| 117 | Erwin Doescher | 2 | 2021 | |||
| 118 | Lélio R.Paula | 1 | 2024 | Brazil |  |
|
Before computer epoch
Factors found by year
| Years | Interval years | Found divisors |
|---|---|---|
| 1640-1731 | 92 | 0 |
| 1732-1854 | 123 | 2 |
| 1855-1900 | 46 | 7 |
| 1901-1952 | 52 | 7 |
| TOTAL | 313 | 16 |
16 factors found earlier (1732-1925)
1640 Dec 25
Fermat wrote to Mersenne:\ABIf I can determine the basic reason why
3, 5, 17, 257, 65 537, ...
are prime numbers, I feel that I would find very interesting results,
for I have already found marvelous things [along these lines] which I will tell you about later.\BB Thus Fermat did not know, that F5 was not prime
[Knuth, The Art Of Computer Programming, vol. 2, #4.5.4]
Neither of them ever resolved this problem, althogh they could have done it as follows:
The number 32^32 mod(232+1) can be computed by doing 32 operations of squaring modulo 232+1, and the answer is 3029026160, therefore (by
Fermat's own theorem, which he discovered in the same year 1640!) the number F5 is not prime.
1730
Goldbach called Euler's attention to Fermat's conjecture that Fm is always prime, and remarked that:
- no Fm has a factor < 100 - no two Fm have a common factor
1732
Euler proved that every factor of Fm is of the form k.2m+1+1,
and noted that each factor of F5 has the form 64k+1, k=10 giving the factor 641: F5=232+1=641*6700417
1801
Gauss in his Disquisitions Arithmaticae, Proved that a regular n-gon can be constructed by ruler and compass
iff n is a product of a power of 2 and distinct odd Fm primes.
1855 Jan 1
Thomas Clausen in a letter to Gauss provided the factorization F6 = 274177 * 67280421310721 with both factors known to be primes, without proving, however,
that the second factor is prime. BUT THIS FACT WAS NEVER PUBLISHED UNTIL 1964 by K.Biermann.
1877
T.Pepin: (basic idea from Lucas) Fm is prime (m>1) iff a(Fm-1)/2 = -1 mod Fm
where a is any non-residue of Fm like 3, 5 or 10
1877 Nov.
Pervouchin: 114689 = 7.214 +1 divides F12. Lucas announced the same result 2 months later.
1878 Jan 27.
Lucas: every factor of Fm is of the form k.2m+2 +1. Click here for detail information.
1878 Feb.
Pervouchin: 167772161 = 5.225 +1 divides F23
1878
Proth: Let N=k.2n+1, where k<2n is odd. Suppose that (a/N)=-1. Then N is prime iff a(N-1)/2 = -1 mod N
1879
Lucas: had verified the compositeness of F6 in 30 hours.
1880 July 7
Landry, when of age 82 and after several months labor found that F6 = 274177 * 67280421310721
both factors being prime, the second with some doubt. See Landry's mail to Lucas.
1880
LeLasseur and Gerardin: each verified the primeness of the second factor of F6
1886
Seelhoff: 2748779069441 = 5.239 +1 divides F36
1896
Jacques Hadamard: found a very simple proof of the 1878 Lucas's theorem.
1899
Cunningham: found two factors of F11 by trial division: 319489 = 39.213 +1, 974849 = 119.213 +1
1903
Western:
2424833 = 37.216 +1 divides F9
13631489 = 13.220 +1 divides F18
26017793 = 397.216 +1 divides F12
63766529 = 973.216 +1 divides F12
1903
Cullen & Cunningham: 6597069766657 = 3.241 +1 divides F38 Western verified its primality
1903
Western & Cunningham: no more Fm have factors < 10^6
1905
Morehead & Western (independently) using Pepin's test with a=3 verified that F7 is composite
1905
Morehead: 188894659314785808547841 = 5.275+1 divides F73
1909
&Morehead & Western (by a very long computation) verified that F8 is composite. (Pepin test a=3)
1925
Kraitchik
1214251009 = 579.221+1 divides F15
Computer epoch
Beginnings from the famous computer SWAC
| Year | Divisors | Year | Divisors | Year | Divisors | Year | Divisors | Year | Divisors | Year | Divisors | Year | Divisors | Year | Divisors |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| - | - | 1961 | - | 1971 | - | 1981 | 3 | 1991 | 12 | 2001 | 22 | 2011 | 9 | 2021 | 5 |
| - | - | 1962 | 2 | 1972 | - | 1982 | 2 | 1992 | 10 | 2002 | 8 | 2012 | 16 | 2022 | - |
| 1953 | 2 | 1963 | 11 | 1973 | - | 1983 | 2 | 1993 | 10 | 2003 | 8 | 2013 | 7 | 2023 | 8 |
| 1954 | - | 1964 | - | 1974 | 2 | 1984 | 7 | 1994 | 1 | 2004 | 2 | 2014 | 7 | 2024 | 3 |
| 1955 | - | 1965 | - | 1975 | - | 1985 | 2 | 1995 | 8 | 2005 | 7 | 2015 | 6 | ||
| 1956 | 14 | 1966 | - | 1976 | 2 | 1986 | 12 | 1996 | 7 | 2006 | 1 | 2016 | 7 | ||
| 1957 | 6 | 1967 | - | 1977 | 4 | 1987 | 5 | 1997 | 4 | 2007 | 4 | 2017 | 5 | ||
| 1958 | - | 1968 | - | 1978 | 2 | 1988 | 4 | 1998 | 8 | 2008 | 6 | 2018 | 7 | ||
| 1959 | - | 1969 | - | 1979 | 13 | 1989 | - | 1999 | 9 | 2009 | 6 | 2019 | 3 | ||
| 1960 | - | 1970 | 2 | 1980 | 9 | 1990 | 8 | 2000 | 13 | 2010 | 7 | 2020 | 5 | ||
| TOTAL | 22 | TOTAL | 15 | TOTAL | 32 | TOTAL | 45 | TOTAL | 82 | TOTAL | 71 | TOTAL | 72 | TOTAL | 16 |
355 prime factors found since the advent of computers
371 prime factors currently known
Thank you to John Bellingham from Sydney, Australia, who spotted and corrected an error on the page.Thank you to Lélio Ribeiro de Paula from Brazil, who shared his historical researches with us.