有一次我在火车上碰见了一位数学家，他对Paul Erdos和他的数学表示难以置信。
他很难想象，一位像Jean-Pierre Serre这样的数学家竟然也会有一个有限的Erdos数。
--Peter J.Cameron
Erdos研究的领域主要是数论和组合数学，但他的论文中涵盖的学科有逼近论、初等几
何、集合论、概率论、数理逻辑、格与序代数结构、线性代数、群论、拓扑群、多项式
、测度论、单复变函数、差分方程与函数方程、数列、Fourier分析、泛函分析、一般
拓扑和代数拓扑、统计、数值分析、计算机科学、信息论等等。"Mathematical Review
s"曾把数学划分为大约六十个分支，Erdos的论文涉及到了其中的40%.
如此广泛的研究方向，再加上他的惊人数量的论文，超长的合作者名单，使他的触角伸展
到了数学的几乎每一个角落。
几乎每一个当代数学家都有一个有限的Erdos数，而且这个数往往非常小，小得出乎本人
的预料。比如说证明Fermat大定理的Andrew Wiles，他的研究方向与Erdos相去甚远，但
他的Erdos数只有3，是通过这个途径实现的：Erdos--Andrew Odlyzko--Chris M.Skinner
--Andrew Wiles.
Fields奖得主的Erdos数都不超过5 （只有Cohen和Grothendieck的Erdos数是5），
Nevanlinna奖得主的Erdos数不超过3 （只有Valiant的Erdos数是3），Wolf数学奖得主
的Erdos数不超过6 （只有V.I.Arnold是6，且只有Kolmogorov是5），Steele奖的终身
成就奖得主的Erdos数不超过4.
在具有有限Erdos数的人名单中往往还能发现一些其他领域的专家，比如Bill Gates，他
的Erdos数是4，通过如下途径实现：Erdos--Pavol Hell--Xiao Tie Deng (邓小铁)--
Christos H. Papadimitriou--William H. (Bill) Gates.
注：
Jean-Pierre Serre，法国数学家，1954年获Fields奖，2000年获Wolf奖。
Pierre de Fermat，17世纪法国数学家，被誉为"业余数学家之王"。但他的数学水平完全
是职业水准的，E.T.Bell甚至说他是17世纪最出色的数学家。
Andrew Wiles，英国数学家。Fermat大定理的证明给他带来了无数的荣誉，其中包括1996
年的Wolf奖和1998年国际数学家大会上颁发的特别奖。
John Charles Fields，加拿大数学家。他是1924年在多伦多举行的国际数学家大会的主席。
以他名字命名的著名奖项被认为是数学界的最高荣誉。
Alexander Grothendieck，法国数学家，现代抽象代数几何的集大成者，1966年获Fields
奖。
Rolf Herman Nevanlinna，芬兰数学家。国际数学家大会上设立有以他名字命名的奖项，颁
发给对信息科学有重大贡献的数学家。
Vladimir Igorevich Arnold，苏联/俄罗斯数学家，主要研究常微分方程与动力系统。1982
年获首届Crafoord奖，2001年获Wolf奖。
Andrey Nikolaevich Kolmogorov，20世纪最著名、最重要的数学家之一，苏联数学界的领
袖。1980年获Wolf奖。
附录：
下面是一些著名学者的Erdos数，其中某些Erdos数小得出乎你的想象之外。
(a) Nobel Prize winners
NAME YEAR SUBJECT ERDOS NUMBER
Max von Laue 1914 Physics 4
Albert Einstein 1921 Physics 2
Niels Bohr 1922 Physics 5
Louis de Broglie 1929 Physics 5
Werner Heisenberg 1932 Physics 4
Paul A. Dirac 1933 Physics 4
Erwin Schrodinger 1933 Physics 8
Enrico Fermi 1938 Physics 3
Ernest O. Lawrence 1939 Physics 6
Otto Stern 1943 Physics 3
Isidor I. Rabi 1944 Physics 4
Wolfgang Pauli 1945 Physics 3
Frits Zernike 1953 Physics 6
Max Born 1954 Physics 3
Willis E. Lamb 1955 Physics 3
John Bardeen 1956 Physics 5
Walter H. Brattain 1956 Physics 6
William B. Shockley 1956 Physics 6
Chen Ning Yang 1957 Physics 4
Tsung-dao Lee 1957 Physics 5
Emilio Segrè 1959 Physics 4
Owen Chamberlain 1959 Physics 5
Robert Hofstadter 1961 Physics 5
Eugene Wigner 1963 Physics 4
Richard P. Feynman 1965 Physics 4
Julian S. Schwinger 1965 Physics 4
Hans A. Bethe 1967 Physics 4
Luis W. Alvarez 1968 Physics 6
Murray Gell-Mann 1969 Physics 3
John Bardeen 1972 Physics 5
Leon N. Cooper 1972 Physics 6
John R. Schrieffer 1972 Physics 5
Aage Bohr 1975 Physics 5
Ben Mottelson 1975 Physics 5
Leo J. Rainwater 1975 Physics 7
Steven Weinberg 1979 Physics 4
Sheldon Lee Glashow 1979 Physics 2
Abdus Salam 1979 Physics 3
S. Chandrasekhar 1983 Physics 4
Norman F. Ramsey 1989 Physics 3
Paul A. Samuelson 1970 Economics 5
Kenneth J. Arrow 1972 Economics 3
Tjalling C. Koopmans 1975 Economics 4
Herbert A. Simon 1978 Economics 3
Gerard Debreu 1983 Economics 3
Franco Modigliani 1985 Economics 4
Robert M. Solow 1987 Economics 4
Harry M. Markowitz 1990 Economics 2
Merton H. Miller 1990 Economics 4
John C. Harsanyi 1994 Economics 8
John F. Nash 1994 Economics 4
Reinhard Selten 1994 Economics 7
Robert C. Merton 1997 Economics 6
Peter J. Debye 1936 Chemistry 5
George De Hevesy 1943 Chemistry 7
Otto Diels 1950 Chemistry 7
Kurt Alder 1950 Chemistry 6
Edwin M. McMillan 1951 Chemistry 6
Glenn T. Seaborg 1951 Chemistry 5
Linus Pauling* 1954 Chemistry 4
*Also received the 1962 Nobel Peace Prize
Lars Onsager 1968 Chemistry 3
Ilya Prigogine 1977 Chemistry 6
Walter Gilbert 1980 Chemistry 4
Kenichi Fukui 1981 Chemistry 3
Jerome Karle 1985 Chemistry 4
Herbert A. Hauptman 1985 Chemistry 3
Francis H. C. Crick 1962 Medicine 7
James D. Watson 1962 Medicine 8
Sir John Carew Eccles 1963 Medicine 3
(b) Fields Medal winners
NAME YEAR COUNTRY ERDOS NUMBER
Lars Ahlfors 1936 Finland 4
Jesse Douglas 1936 USA 4
Laurent Schwartz 1950 France 4
Atle Selberg 1950 Norway 2
Kunihiko Kodaira 1954 Japan 2
Jean-Pierre Serre 1954 France 3
Klaus Roth 1958 Germany 2
Rene Thom 1958 France 4
Lars Hormander 1962 Sweden 3
John Milnor 1962 USA 3
Michael Atiyah 1966 Great Britain 4
Paul Cohen 1966 USA 5
Alexander Grothendieck 1966 Germany/France 5
Stephen Smale 1966 USA 4
Alan Baker 1970 Great Britain 2
Heisuke Hironaka 1970 Japan 4
Serge Novikov 1970 USSR 3
John G. Thompson 1970 USA 3
Enrico Bombieri 1974 Italy 2
David Mumford 1974 Great Britain 2
Pierre Deligne 1978 Belgium 3
Charles Fefferman 1978 USA 2
Gregori Margulis 1978 USSR 4
Daniel Quillen 1978 USA 3
Alain Connes 1982 France 3
William Thurston 1982 USA 3
Shing-Tung Yau 1982 China 2
Simon Donaldson 1986 Great Britain 4
Gerd Faltings 1986 Germany 4
Michael Freedman 1986 USA 3
Valdimir Drinfeld 1990 USSR 4
Vaughan Jones 1990 New Zealand 4
Shigemufi Mori 1990 Japan 3
Edward Witten 1990 USA 3
Pierre-Louis Lions 1994 France 4
Jean Christophe Yoccoz 1994 France 3
Jean Bourgain 1994 Belgium 2
Efim Zelmanov 1994 Russia 3
Richard Borcherds 1998 S Afr/Gt Brtn 2
William T. Gowers 1998 Great Britain 4
Maxim L. Kontsevich 1998 Russia 4
Curtis McMullen 1998 USA 3
(c) Nevanlinna Prize winners
NAME YEAR COUNTRY ERDOS NUMBER
Robert Tarjan 1982 USA 2
Leslie Valiant 1986 Hungary/Gt Brtn 3
Alexander Razborov 1990 Russia 2
Avi Wigderson 1994 Israel 2
Peter Shor 1998 USA 2
(d) Wolf Prize in Mathematics winners
NAME YEAR COUNTRY ERDOS NUMBER
Izrail M. Gelfand 1978 USSR (Russia) 4
Carl L. Siegel 1978 Germany 3
Jean Leray 1979 France 3
André Weil 1979 France 4
Henri Cartan 1980 France 3
Andrei N. Kolmogorov 1980 Russia 5
Lars Ahlfors 1981 Finland 4
Oscar Zariski 1981 USA 3
Hassler Whitney 1982 USA 2
Mark G. Krein 1982 Ukranian SSR 4
Shiing Shen Chern 1983-84 China 2
Paul Erd?s 1983-84 Hungary 0
Kunihiko Kodaira 1984-85 Japan 2
Hans Lewy 1984-85 Germany 3
Samuel Eilenberg 1986 Poland 2
Atle Selberg 1986 Norway 2
Kiyoshi Ito 1987 Japan 3
Peter D. Lax 1987 Hungary/USA 3
Friedrich E. Hirzebruch 1988 Germany 3
Lars Hormander 1988 Sweden 3
Alberto Calderón 1989 Argentina 3
John Milnor 1989 USA 3
Ennio De Giorgi 1990 Italy 3
Ilya Piatetski-Shapiro 1990 Russia 3
Lennart A. Carleson 1992 Sweden 4
John G. Thompson 1992 USA 3
Mikhael Gromov 1993 Russia 3
Jacques Tits 1993 Belgium 4
Jurgen K. Moser 1994-95 Germany 3
Robert Langlands 1995-96 Canada 2
Andrew Wiles 1995-96 Great Britain 3
Joseph B. Keller 1997 USA 3
Yakov G. Sinai 1997 Russia 4
Laszlo Lovasz 1999 Hungary 1
Elias M. Stein 1999 Belgium/USA 2
Raoul Bott 2000 Hungary 3
Jean-Pierre Serre 2000 France 3
Vladimir I. Arnold 2001 Russia 6
Saharon Shelah 2001 Israel 1
(e) Steele Prize (Lifetime Achievement) winners
NAME YEAR COUNTRY ERDOS NUMBER
Salomon Bochner 1979 Poland 2
Antoni Zygmund 1979 Poland 2
André Weil 1980 France 4
Gerhard P. Hochschild 1980 Germany 4
Oscar Zariski 1981 Poland 3
Fritz John 1982 Germany 4
Shiing Shen Chern 1983 China 2
Joseph L. Doob 1984 USA 2
Hassler Whitney 1985 USA 2
Saunders Mac Lane 1986 USA 3
Samuel Eilenberg 1987 Poland 2
Deane Montgomery 1988 USA 3
Raoul Bott 1990 Hungary 3
Armand Borel 1991 Switzerland 3
Peter D. Lax 1992 Hungary/USA 3
Eugene B. Dynkin 1993 Russia/USA 3
Louis Nirenberg 1994 Canada 3
John T. Tate 1995 USA 3
Goro Shimura 1996 Japan 2
Ralph S. Phillips 1997 USA 2
Nathan Jacobson 1998 USA 3
Richard V. Kadison 1999 USA 3
Isadore M. Singer 2000 USA 3
Harry Kesten 2001 USA 2
(f) Other distinguished scholars
NAME FIELD OF RESEARCH ERDOS NUMBER
Walter Alvarez geology 7
Rudolf Carnap philosophy 4
Jule G. Charney meteorology 4
Noam Chomsky linguistics 4
Freeman J. Dyson quantum physics 2
George Gamow nuclear physics and cosmology 5
Stephen Hawking relativity and cosmology 4
John L. Hennessey comp. sci., Stanford U president 4
Pascual Jordan quantum physics 4
Theodore von Kármán aeronautical engineering 4
John Maynard Smith biology 4
Oskar Morgenstern economics 4
Rolf Nevanlinna mathematics 5
J. Robert Oppenheimer nuclear physics 4
Roger Penrose relativity and cosmology 3
Jean Piaget psychology 3
Karl Popper philosophy 4
Edwin E. Salpeter astrophysics 5
Claude E. Shannon electrical engineering 3
Arnold Sommerfeld atomic physics 5
Edward Teller nuclear physics 4
Alan Turing computer science 5
George Uhlenbeck atomic physics 2
John A. Wheeler nuclear physics 3
