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History

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


 Computer calculations

13 John L. Selfridge (1927-2010) 6 1953 USA
14 Raphael Mitchel Robinson (1911-1994) 20 1956 USA
15 John Brillhart (1930 - ) 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 - ) 78 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 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 1 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 6 2012 USA
106 Ryan Propper 1 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

Before computer epoch

Factors found by year

YearsInterval yearsFound divisors
1640-1731920
1732-18541232
1855-1900467
1901-1952527
TOTAL31316

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
- - 1961 - 1971 - 1981  3 1991 12 2001 22 2011  9
- - 1962  2 1972 - 1982  2 1992 10 2002  8 2012 16
1953  2 1963 11 1973 - 1983  2 1993 10 2003  8 2013  7
1954 - 1964 - 1974  2 1984  7 1994  1 2004  2 2014  7
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  3
1959 - 1969 - 1979 13 1989 - 1999  9 2009  6
1960 - 1970  2 1980  9 1990  8 2000 13 2010  7
TOTAL 22 TOTAL 15 TOTAL 32 TOTAL 45 TOTAL 82 TOTAL 71 TOTAL 60

327 prime factors found since the advent of computers

343 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.

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