**Paul Erdős** (March 26, 1913 - September 20, 1996) was an immensely prolific mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, and number theory.

He was born in Budapest, Hungary as *Erdős Pál* (pronounced *"Air-dersh"*, SAMPA /errd9:S/), into a non-practicing Jewish family. The Budapest Jewish community of that day produced at least four remarkable thinkers besides Erdős: Eugene Wigner, the physicist and engineer; Edward Teller, the physicist and politician; Leo Szilard, the chemist, physicist and politician; and John von Neumann, the mathematician and Renaissance man.

## Early life

Erdős was an only child, having lost his only siblings (two sisters aged three and five) to scarlet fever just days before he was born. Both his parents taught mathematics, and Erdős, by the age of four, had independently observed several established properties of prime numbers.

In 1914, Erdős' father Lajos was captured by the Russian army in its attack on the Austro-Hungarian alliance. He spent six years in captivity in Siberia. Erdős' mother Anna, excessively protective after the loss of her husband and two daughters, kept Paul away from school for much of his early years but hired a tutor to teach him at home. In 1920, Lajos Erdős returned from captivity and continued the education of his son in mathematics and English.

Despite the restrictions on Jews entering universities in Hungary, Erdős was accepted in 1930, having won a national examination. In 1934, he was awarded his doctorate. Anxious about rising anti-semitism in Hungary during the 1930s, he took a post-doctoral fellowship at Manchester. In 1938, he took his first American position at Princeton, though he did not get tenure because the administration of Princeton deemed him "uncouth and unconventional". Around this time Erdős began his habit of travelling from campus to campus which was to define his professional career.

All accounts report Erdős to have been a naďve, almost childlike character. An awkward incident occurred in 1941 when Erdős and another mathematician became involved in a heated discussion about a point of mathematical theory, and failed to notice they were too near a military communication facility on Long Island. They were arrested for trespassing and on suspicion of spying, and Erdős unfortunately earned an FBI record.

## Contributions to mathematics

Erdős' characteristic style was to solve complicated problems in an elegant and visionary manner. He received the *Cole Prize* of the *American Mathematical Society* in 1951 for his many papers on the theory of numbers, and in particular for the paper "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem", published in the Proceedings of the National Academy of Sciences in 1949. Erdős is also known for the unproven Erdős conjecture in graph theory, which claims that any graph with minimum degree 3 contains a cycle whose length is a power of 2.

## Exile

In the early 1950s Erdős' FBI record brought him to the attention of the McCarthy Investigations, and Erdős suddenly found he could not obtain a visa to the United States. Consequently, he spent much of the next ten years in Israel. During the early 1960s he made numerous requests to be allowed to return to the United States and a visa was finally granted in November 1963.

For the next thirty years Erdős "officially" held posts at a number of universities in Israel, the US and Great Britain, but in reality he was a permanent nomad of no fixed agenda or location, wandering to all major universities as he saw fit.

His working habits became characterised by working obsessively long hours, sleeping at most 4-5 hours a night and relying on the heavy use of amphetamines to maintain his activity levels. In 1979, his friend Ronald Graham offered a $500 bet, challenging him to go without drugs for 30 days. Erdős met the challenge, but later complained bitterly that the progress of mathematics had been held up for a month by this silly bet.

## Mathematical collaboration

Erdős' prestige and genius usually guaranteed him a warm welcome at any universities, and he inevitably completed a paper with any mathematician who could present a topic that appealed to him. As a result, he may be the most collaborative mathematician ever, with nearly 1,500 jointly published papers. The community of mathematicians who worked with him created the (tongue in cheek) Erdős number in his honor.

Because he rarely published alone, Erdős more than anyone else was credited with "turning mathematics into a social activity". Among his frequent collaborators were Yousef Alavi, Bela Bollobas, Fan Chung, Ralph Faudree, Ron Graham, Andras Gyarfas, Andras Hajnal, Eric Milner, Janos Pach, Carl Pomerance, Richard Rado (one of the co-authors of the famous Erdős-Ko-Rado theorem), Alfred Renyi, Vojtech Rődl, Cecil C. Rousseau, Andras Sarkozy, Dick Schelp, Miklos Simonovits, Vera Sos, Joel Spencer, Endre Szemeredi, Pal Turan and Peter Winkler.

Erdős was a constant source of witty aphorisms: "Another roof, another proof", "A mathematician is a machine for turning coffee into theorems" (a saying sometimes attributed to Alfred Renyi), "You don't have to believe in God, but you should believe in The Book" (where *The Book* was a hypothetical divinely held volume containing the most succinct, elegant and illuminating proofs for all mathematical statements). Erdős used the phrase "to leave" for people who died, and the phrase "to die" for people who stopped doing mathematics. He called children epsilons and was fond of them.

Erdős went on to receive many awards including the Wolf Prize of $50,000 in 1983. He lived simply, and gave most of his money to help favored students or as prizes for solving problems he had posed.

He died in Warsaw a few hours after a conference.

*See also: Erdős-Borwein constant*

## External links and references

- O'Connor and Robertson, "Erdős", MacTutor biography
- Paul Hoffman, "The man who loved only numbers", Hyperion 1998. Biography. ISBN 0786863625.
- Bruce Schechter, "My brain is open", Touchstone 2000. Biography. ISBN 0684859807