有一次我在火車上碰見了一位數學家，他對Paul Erdos和他的數學表示難以置信。
他很難想象，一位像Jean-Pierre Serre這樣的數學家竟然也會有一個有限的Erdos數。
--Peter J.Cameron
Erdos研究的領域主要是數論和組合數學，但他的論文中涵蓋的學科有逼近論、初等幾
何、集合論、概率論、數理邏輯、格與序代數結構、線性代數、群論、拓撲群、多項式
、測度論、單複變函數、差分方程與函數方程、數列、Fourier分析、泛函分析、一般
拓撲和代數拓撲、統計、數值分析、計算機科學、信息論等等。"Mathematical Review
s"曾把數學劃分為大約六十個分支，Erdos的論文涉及到了其中的40%.
如此廣泛的研究方向，再加上他的驚人數量的論文，超長的合作者名單，使他的觸角伸展
到了數學的幾乎每一個角落。
幾乎每一個當代數學家都有一個有限的Erdos數，而且這個數往往非常小，小得出乎本人
的預料。比如說證明Fermat大定理的Andrew Wiles，他的研究方向與Erdos相去甚遠，但
他的Erdos數只有3，是通過這個途徑實現的：Erdos--Andrew Odlyzko--Chris M.Skinner
--Andrew Wiles.
Fields獎得主的Erdos數都不超過5 （只有Cohen和Grothendieck的Erdos數是5），
Nevanlinna獎得主的Erdos數不超過3 （只有Valiant的Erdos數是3），Wolf數學獎得主
的Erdos數不超過6 （只有V.I.Arnold是6，且只有Kolmogorov是5），Steele獎的終身
成就獎得主的Erdos數不超過4.
在具有有限Erdos數的人名單中往往還能發現一些其他領域的專家，比如Bill Gates，他
的Erdos數是4，通過如下途徑實現：Erdos--Pavol Hell--Xiao Tie Deng (鄧小鐵)--
Christos H. Papadimitriou--William H. (Bill) Gates.
註：
Jean-Pierre Serre，法國數學家，1954年獲Fields獎，2000年獲Wolf獎。
Pierre de Fermat，17世紀法國數學家，被譽為"業餘數學家之王"。但他的數學水平完全
是職業水準的，E.T.Bell甚至說他是17世紀最出色的數學家。
Andrew Wiles，英國數學家。Fermat大定理的證明給他帶來了無數的榮譽，其中包括1996
年的Wolf獎和1998年國際數學家大會上頒發的特別獎。
John Charles Fields，加拿大數學家。他是1924年在多倫多舉行的國際數學家大會的主席。
以他名字命名的著名獎項被認為是數學界的最高榮譽。
Alexander Grothendieck，法國數學家，現代抽象代數幾何的集大成者，1966年獲Fields
獎。
Rolf Herman Nevanlinna，芬蘭數學家。國際數學家大會上設立有以他名字命名的獎項，頒
發給對信息科學有重大貢獻的數學家。
Vladimir Igorevich Arnold，蘇聯/俄羅斯數學家，主要研究常微分方程與動力系統。1982
年獲首屆Crafoord獎，2001年獲Wolf獎。
Andrey Nikolaevich Kolmogorov，20世紀最著名、最重要的數學家之一，蘇聯數學界的領
袖。1980年獲Wolf獎。
附錄：
下面是一些著名學者的Erdos數，其中某些Erdos數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
