The Erdős number () describes the "collaborative distance" between mathematician Paul Erdős and another person, measured by joint authorship of mathematical papers. The same principle has been applied in other fields.
Overview
Paul Erdős (1913–1996) was an influential Hungarian mathematician who, in the latter part of his life, spent a great deal of time writing papers with a large number of colleagues—more than 500—working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.) Erdős spent most of his career with no permanent home or job. He traveled with everything he owned in two suitcases, and would visit mathematicians with whom he wanted to collaborate, often unexpectedly, and expect to stay with them. File available as HTML via Solipsys
The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy. For example, Erdős collaboration graphs can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.
Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. Original Spanish version in Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 23 (89) 563–582, 1999, . The median Erdős number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the lowest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.
Definition and application in mathematics
To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős himself is assigned an Erdős number of zero. A certain author's Erdős number is one greater than the lowest Erdős number of any of their collaborators; for example, an author who has coauthored a publication with Erdős would have an Erdős number of 1. The American Mathematical Society provides a free online tool to determine the collaboration distance between two mathematical authors listed in the Mathematical Reviews catalogue.
Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (13,782 people as of 10 August 2025Erdos2, Version 2025, August 10, 2025.), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of infinity (or an undefined one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.
There is room for ambiguity over what constitutes a link between two authors. The American Mathematical Society collaboration distance calculator uses data from Mathematical Reviews, which includes most mathematics journals but covers other subjects only in a limited way, and which also includes some non-research publications. The Erdős Number Project web site says: It also says:
but excludes non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The "Erdős number of the second kind" restricts assignment of Erdős numbers to papers with only two collaborators.Grossman et al. "Erdős numbers of the second kind," in Facts about Erdős Numbers and the Collaboration Graph. The Erdős Number Project, Oakland University, USA. Retrieved July 25, 2009.
The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?" See also some comments in an obituary by Michael Golomb.
The median Erdős number among Fields medalists is as low as 3. Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, Alan Baker, Enrico Bombieri, David Mumford, Charles Fefferman, William Thurston, Shing-Tung Yau, Jean Bourgain, Richard Borcherds, Manjul Bhargava, Jean-Pierre Serre and Terence Tao. There are no Fields medalists with Erdős number 1; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.
Most frequent Erdős collaborators While Erdős collaborated with hundreds of co-authors, there were some individuals with whom he co-authored dozens of papers. This is a list of the ten persons who most frequently co-authored with Erdős and their number of papers co-authored with Erdős, i.e., their number of collaborations.Grossman, Jerry, Erdos0p, Version 2010, The Erdős Number Project, Oakland University, US, October 20, 2010. Co-author Number of collaborations András Sárközy 62 András Hajnal 56 Ralph Faudree 50 Richard Schelp 42 Cecil C. Rousseau 35 Vera T. Sós 35 Alfréd Rényi 32 Pál Turán 30 Endre Szemerédi 29 Ronald Graham 28
Related fields , all Fields medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine, and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average, and median Erdős numbers among those laureates.
+ Statistics on Mathematical Collaboration, 1903–2016 #Laureates #Erdős %Erdős Min Max Average MedianFields Medal5656100.0%263.363Nobel Economics764761.84%284.114Nobel Chemistry1724224.42%3105.485Nobel Medicine2105827.62%3125.505Nobel Physics20015979.50%2125.635
Physics Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Glashow have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include Enrico Fermi, Otto Stern, Wolfgang Pauli, Max Born, Willis E. Lamb, Eugene Wigner, Richard P. Feynman, Hans A. Bethe, Murray Gell-Mann, Abdus Salam, Steven Weinberg, Norman F. Ramsey, Frank Wilczek, David Wineland, and Giorgio Parisi. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.
Angela Merkel, Chancellor of Germany from 2005 to 2021, has an Erdős number of at most 5.
Biology Several prolific scientists working in Genetics, Biomedical Engineering, Mathematical, and Computational Biology have an Erdős number of 2. Among them are Zvia Agur, Joel E. Cohen, Eugene Koonin, Bruce Kristal, Eric Lander, Lior Pachter and Temple F. Smith. Through collaborations with these authors there are many biologists with an Erdős number of 3 and it has been argued that almost every author on a paper in the biological sciences can be linked to Erdős.
Finance and economics There are at least two winners of the Nobel Prize in Economics with an Erdős number of 2: Harry M. Markowitz (1990) and Leonid Kantorovich (1975). Other financial mathematicians with Erdős number of 2 include David Donoho, Marc Yor, Henry McKean, Daniel Stroock, and Joseph Keller.
Nobel Prize laureates in Economics with an Erdős number of 3 include Kenneth J. Arrow (1972), Milton Friedman (1976), Herbert A. Simon (1978), Gerard Debreu (1983), John Forbes Nash, Jr. (1994), James Mirrlees (1996), Daniel McFadden (2000), Daniel Kahneman (2002), Robert J. Aumann (2005), Leonid Hurwicz (2007), Roger Myerson (2007), Alvin E. Roth (2012), and Lloyd S. Shapley (2012) and Jean Tirole (2014).Grossman, J. (2015): "The Erdős Number Project." http://wwwp.oakland.edu/enp/erdpaths/
Some investment firms have been founded by mathematicians with low Erdős numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdős number of 3.
Philosophy Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdős numbers are available for many philosophers. Philosophers John P. Burgess and Brian Skyrms have an Erdős number of 2. Jon Barwise and Joel David Hamkins, both with Erdős number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.
Law Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher, and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.
Engineering Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdős. It is therefore not surprising that practitioners in these fields have low Erdős numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdős number of 1, having collaborated with Erdős himself. Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2.
Linguistics The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.
Impact
Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65; almost everyone with a finite Erdős number has a number less than 8.
Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers. For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via Persi Diaconis or Paul Deheuvels, who have Erdős numbers of 1, or John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers, and since Kleitman has an Erdős number of 1, a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.
According to Alex Lopez-Ortiz, all the Fields and Nevanlinna Prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9.