LINKS
CONTACTS



 
History

All researchers of Fermat numbers, who found at least one factor

NComplete name and
links to the biography
or personal sites.
Years of lifeQuantity of
found divisors
personally or
in co-authorship
Year
of the
first
mention
Country Portrait

Manual calculations

1Pierre Fermat(1601-1665)5 prime1640France
2Leonhard Euler(1707-1783)21732Switzerland/Russia
3Thomas Clausen(1801-1885)21855Denmark/Russia
4Edouard Lucas(1842-1891)11877France
5

Ivan Mikheevich Pervushin

(1829-1900)21877Russia
6Fortune Landry(1798-?)21880France
7Paul Peter Heinrich Seelhoff(1829-1896)11886Germany
8Allan J.C. Cunningham31899
9A.E.Western51903
10James Cullen(1867-1933)11903Ireland
11J.C.Morehead11906
12Maurice Borisovich Kraitchik(1882-1957)11925Belgium


 Computer calculations

13John L. Selfridge (1927 - 2010)61953USA
14Raphael M. Robinson(1911-1994)201956USA
15John Brillhart(1930 - )51962USA
16Hans Riesel(1929 - )11962Sweden
17Claude P. Wrathall111963USA
18Michael A.Morrison21970
19John C. Hallyburton21974
20G. Matthew21976
21Hugh C. Williams (1943 - )61976Canada
22D.E.Shippee41977USA
23Gary B.Gostin(1954 - )651978USA
24Wilfrid Keller(1937 - )131978Argentina/Germany
25Robert J. Baillie51979
26Philip B. McLaughlin71979USA
27Hiromi Suyama71979Japan
28A.Oliver L.Atkin(1925-2008)51979England
29N.W.Rickert51979
30Gordon V. Cormack(1956 - )41979Canada
31Richard Peirce Brent(1946 - )81980Australian
32John M.Pollard21980England
33Francois Morain11988France
34Arjen K.Lenstra (1956 - )21990
35Mark S.Manasse21990USA
36Richard E. Crandall(1947 - 2012)51991USA
37Harvey Dubner81992USA
38Jeffrey Young81993USA
39Tadashi Taura(1954 -)181995Japan
40Karl Dilcher (1954 - )11996Germany/Canada
41Chris Van Halewyn11997
42Patrick Demichel11997France
43Yves Gallot(1966 - )281997France
44

Robert Prethaler

11998
45

Gennady Nikolaevich Gusev

(1954 - )11998Russia
46

Richard McIntosh

11999
47

Claude Tardif

11999
48

Charles F. Kerchner III

31999
49

Dan Morenus

11999
50

John Renze

11999
51

John B. Cosgrave

(1946 - )31999Ireland
52

Rachel Lewis

12000USA
53

Leonid Nikolaevich Durman

(1975 - )232000Russia
54James Ray Ballinger(1953 - )12000USA
55

Nestor Sergio de Araujo Melo

(1971 - )22000Brazil
56Payam Samidoost(1969 - )42000Iran
57Takahiro Nohara(1969 - )102001Japan
58Peter Grobstich(1942 - )22001Germany
59

Alexander Kruppa

(1974 - )22001Germany
60

Tony Forbes

(1944 - )12001England
61

Marko Bodschwinna

(1974 - )12001Germany
62

Göran Axelsson

(1966 - )22001Sweden
63

Paul Jobling

(1964 - )202001England
64

George Woltman

(1957 - )262001USA
65

Jim Fougeron

(1966 - )192001USA
66

Vasily Anatol'evich Danilov

(1957 - )52001Russia
67

Anton Petrovich Oleynik

(1972 - )12001Russia
68

Dmitry Komin

(1971 - )12002Russia
69

Kevin Odermatt

12002
70

Sergey Kuzmin

12003Russia
71

Craig Kitchen

12003USA
72

Asko Vuori (1965 - )

22005Finland
73

Maximilian Pacher (1971 - )

52005Austria
74

Michael Eaton

22005USA
75

Jun Tajima

12005Japan
76

Curtis Cooper

22005USA
77

Reto Keiser

12007Switzerland
78

Jean Penné

102007France
79

Pavlos Saridis (1980 - )

12007Greece
80

Souichi Murata

12007Japan
81

Martin Ptáček (1985 - )

12008Czech Republic
82

Eric Ueda

12008USA
83

Senji Yamashita

12009Japan
84

Eric Embling

12009USA
85

David Bessell, GIMPS et al.

32009Australia
86

Sergei Maiorov (1969 - )

12010Russia
87

Tapio Rajala (1982 - ), GIMPS et al.

32010Finland
88

Cedric Vonck (1980 - )

12010Belgium
89

Michael Vang

12010USA
90

Peter Strasser

12010Austria
91

Alexey Komkov

12010Russia
92

Dirk D'huyvetters & PrimeGrid

12011Belgium
93

Roman Maznichenko

52011Russia
94

Scott Brown & PrimeGrid

12011USA
95

Mark Doom & PrimeGrid

12011USA
96

Nikolay Kamenyuk

12011Ukraine
97

Peter Doggart & PrimeGrid

12011England
98

Grzegorz Granowski & PrimeGrid

12011Poland
99

Bruce Dodson & PrimeGrid

12012USA
100

Rob Derrera & PrimeGrid

12012USA
101

Andriy Sen

12012Ukraine
102

James Boerner & PrimeGrid

12012USA
103

Robert Boniecki & PrimeGrid

12012USA
104

Michael Dangler

12012USA
105

Serge Batalov

52012USA
106

Ryan Propper

12013USA
107

Marshall Bishop & PrimeGrid

12013USA
108

Bart van Rooijen & PrimeGrid

12013 -
109

Say Yik Tang & PrimeGrid

12013 -
1110

Raymond Ottusch & PrimeGrid

12014 -

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:«If 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.» 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

DivisorsTOTAL
YearDivisorsYearDivisorsYearYearDivisorsYearDivisorsYearDivisorsYearDivisors
--1961-1971-1981 31991122001222011 9
--1962 21972-1982 21992102002 9201216
1953 21963111973-1983 21993102003 82013 7
1954-1964-1974 21984 71994 12004 22014 4
1955-1965-1975-1985 21995 82005 7
1956141966-1976 21986121996 72006 1
1957 61967-1977 41987 51997 42007 4
1958-1968-1978 21988 41998 82008 6
1959-1969-1979131989-1999 92009 6
1960-1970 21980 91990 82000132010 6
TOTAL2215TOTAL32TOTAL45TOTAL82TOTAL71TOTAL36

303 prime factors found since the advent of computers

319 prime factors currently known

Thank you to John Bellingham from Sydney, Australia, who spotted and corrected an error on the page.

Copyright © MoreWare 2003 ...