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Most famous mathematicians of the past worked largely alone. Paul Erdős did the opposite. He travelled from one university and one friend's home to the next, and wrote papers with hundreds of people. The links were so many that mathematicians invented a number to measure how close they are to him. This post covers who Erdős was, the kind of problems he loved, how Erdős numbers work, a worked example you can do on paper, and what his habits suggest for your own study.
Who was Paul Erdős?
His MacTutor biography gives his birth in Budapest on 26 March 1913 and his death in Warsaw on 20 September 1996. Both his parents taught mathematics. Despite restrictions on Jewish students, he entered university in 1930 as the winner of a national examination, and he earned his doctorate in 1934.
Anti-Jewish laws pushed him abroad, first to a fellowship in Manchester and in 1938 to Princeton. From then on he had no fixed home for long. He moved between universities and the homes of collaborators; Ronald Graham kept a room for him and stored his papers. He won the Wolf Prize in 1983, and MacTutor notes that he gave away most of the money he earned, to help students or as prizes for problems he posed.
What kind of mathematics did he do?
MacTutor calls him a solver of problems rather than a builder of theories, drawn to combinatorics, graph theory and number theory. He wanted proofs that were elementary and elegant, and that showed why a result was true.
A good example is Bertrand's postulate: for every n ≥ 2 there is a prime between n and 2n. Bertrand conjectured it in 1845 and Chebyshev proved it in 1850, but at 18 Erdős found a short elementary proof. In 1949 he and Atle Selberg found an elementary proof of the prime number theorem, and Erdős's paper On a New Method in Elementary Number Theory Which Leads to An Elementary Proof of the Prime Number Theorem helped win him the American Mathematical Society's Cole Prize in 1951.
You can test Bertrand's postulate yourself. Between 10 and 20 lie 11, 13, 17 and 19. Between 25 and 50 lie 29, 31, 37, 41, 43 and 47. Checking cases builds trust, but only a proof covers every n.
What is an Erdős number?
The Erdős Number Project at Oakland University, run by Jerry Grossman, gives the definition. Erdős has Erdős number 0. His coauthors have 1. People who wrote a paper with someone of Erdős number 1, but not with Erdős, have 2, and so on. If no chain links you to him, your number is infinite. The project traces the first published mention to a 1969 article by Casper Goffman in the American Mathematical Monthly.
Its data files page lists 514 people with Erdős number 1 and 13,782 with Erdős number 2. Using 2004 Mathematical Reviews data, the project's facts page puts the median Erdős number at 5 and the mean at 4.65. The same page notes that the share of papers with a single author fell from over 90% in the 1940s to under half.
Worked example: finding Erdős numbers
Sample names, not real people. Suppose these pairs have written joint papers: Erdős and Asha, Erdős and Ben, Asha and Chitra, Ben and Chitra, Chitra and Dev, Dev and Esha, and separately Farah and Gopal.
Number 0. Erdős.
Number 1. Everyone who wrote with Erdős: Asha and Ben.
Number 2. New people who wrote with Asha or Ben: Chitra. She has two routes, through Asha and through Ben, but both have length 2.
Number 3. New people linked to Chitra: Dev.
Number 4. Esha, through Dev.
Infinite. Farah and Gopal wrote only with each other, so no chain reaches Erdős.
Working outwards one layer at a time like this is called breadth-first search. It finds the shortest chain to every person.
What mistakes do students make?
Taking the longest or first path found. Always use the shortest chain.
Counting papers instead of people. Ten joint papers still make one link.
Treating a few checks as a proof. The Bertrand cases above are evidence, not a proof.
Thinking elementary means easy. MacTutor describes his problems as simple to understand but notoriously difficult to solve.
If olympiad number theory is where you want to use ideas like these, our post on IOQM number theory questions is a good next read. When a proof step loses you, the Fermi Problem Library has a step-by-step solver that explains the reasoning.
What can a student take from how Erdős worked?
Work with others. Explaining a problem to someone else is often how it gets solved.
Collect and share problems. He posed problems widely and offered his own money as prizes.
Look for the proof that explains. A short argument that shows why something is true is worth more than a long one that only shows that it is.
Next step: list the primes between 50 and 100, then find and read a short proof of Bertrand's postulate with a friend.
