In 1934, Erdős took up a
post-doctoral fellowship at
Victoria University of Manchester in
Manchester, England, where he met
G. H. Hardy and
Stanisław Ulam.
Time magazine called him "The Oddball's Oddball". Except for some years in the 1950s, when he was not allowed to enter the United States based on the accusation that he was a Communist sympathizer, his life was a continuous series of going from one meeting or seminar to another. In 1943 Erdős worked at
Purdue University in
Indianapolis. In the same year, Stanisław Ulam invited Erdős to work on the
Manhattan Project in
Los Alamos, New Mexico, with him, along with other mathematicians and physicists. However, Erdős expressed a desire to return to Hungary after the war. in 1954, the
United States Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States. The official reasons were the fact that he had corresponded with a Chinese mathematician who had subsequently returned from the United States to China, and also Erdős's 1941 FBI record. In 1985, he visited two universities in
Adelaide,
South Australia,
Flinders University and the
University of Adelaide. At the latter, he met budding mathematician
Terence Tao, then 10 years old. Erdős reportedly enjoyed working with children. His trip to Australia was instigated by longtime friend and collaborator, Hungarian mathematician
George Szekeres.
Mathematical work Erdős was one of the most prolific mathematician and producer of mathematical
conjectures in history, if not the most. He was compared with
Leonard Euler for the sheer quantity of his writings. Erdős wrote around 1,525 mathematical articles in his lifetime – a figure that remained unsurpassed as of 2023 – mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, Most of his work centered on
discrete mathematics,
graph theory,
number theory,
mathematical analysis,
approximation theory,
set theory, and
probability theory. In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics" by
Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated).
Joel Spencer states that "his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular
theorems and conjectures throughout his illustrious career." Erdős never won the
Fields Medal (the highest mathematical prize available during his lifetime), nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the 1983/84
Wolf Prize, "for his numerous contributions to
number theory,
combinatorics,
probability,
set theory and
mathematical analysis, and for personally stimulating mathematicians the world over". Of his contributions, the development of
Ramsey theory and the application of the
probabilistic method especially stand out.
Extremal combinatorics owes to him a whole approach, derived in part from the tradition of
analytic number theory. Erdős found a proof for
Bertrand's postulate which proved to be far neater than
Chebyshev's original one. He also discovered
the first elementary proof for the
prime number theorem, along with
Atle Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg. Erdős also contributed to fields in which he had little real interest, such as
topology, where he is credited as the first person to give an example of a
totally disconnected topological space that is not
zero-dimensional, the
Erdős space.
Erdős's problems , who was 10 years old at the time (1985). Erdős had a reputation for posing new problems as well as solving existing ones:
Ernst Strauss called him "the absolute monarch of problem posers". Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from $25 for problems that he felt were just out of the reach of the current mathematical thinking (both his and others) up to $10,000 for problems that were both difficult to attack and mathematically significant. Some of these problems have since been solved, including the most lucrative – Erdős's conjecture on
prime gaps was solved in 2014, and the $10,000 paid. There are thought to be at least a thousand remaining unsolved problems, though there is no official or comprehensive list. The offers remained active despite Erdős's death;
Ronald Graham was the (informal) administrator of solutions, and a solver could receive either an original check signed by Erdős before his death (for memento only, cannot be cashed) or a cashable check from Graham. Graham's role as administrator was later taken on by the Combinatorics Foundation, currently chaired by
Steve Butler. British mathematician
Thomas Bloom started a website dedicated to Erdős's problems in 2024. Perhaps the most mathematically notable of these problems is the
Erdős conjecture on arithmetic progressions: If true, it would solve several other open problems in number theory, although one main implication of the conjecture, that the
prime numbers contain arbitrarily long arithmetic progressions, has since been proved independently as the
Green–Tao theorem. The payment for the solution of the problem is currently worth US$5,000. The most familiar problem with an Erdős prize is likely the
Collatz conjecture, also called the 3
N + 1 problem. Erdős offered $500 for a solution.
Collaborators Erdős's most frequent collaborators include Hungarian mathematicians
András Sárközy (62 papers) and
András Hajnal (56 papers), and American mathematician
Ralph Faudree (50 papers). Other frequent collaborators were the following: •
Richard Schelp (42 papers) •
Cecil C. Rousseau (35 papers) •
Vera T. Sós (35 papers) •
Alfréd Rényi (32 papers) •
Pál Turán (30 papers) •
Endre Szemerédi (29 papers) •
Ron Graham (28 papers) •
Stefan Burr (27 papers) •
Carl Pomerance (23 papers) •
Joel Spencer (23 papers) •
János Pach (21 papers) •
Miklós Simonovits (21 papers) •
Ernst G. Straus (20 papers) •
Melvyn B. Nathanson (19 papers) •
Jean-Louis Nicolas (19 papers) •
Richard Rado (18 papers) •
Béla Bollobás (18 papers) •
Eric Charles Milner (15 papers) •
András Gyárfás (15 papers) •
John Selfridge (14 papers) •
Fan Chung (14 papers) • Richard R. Hall (14 papers) •
George Piranian (14 papers) • István Joó (12 papers) • Zsolt Tuza (12 papers) • A. R. Reddy (11 papers) •
Vojtěch Rödl (11 papers) •
Pál Révész (10 papers) •
Zoltán Füredi (10 papers) For other co-authors of Erdős, see the list of people with Erdős number 1 in
List of people by Erdős number.
Doctoral students Erdős's doctoral students included: •
Joseph Kruskal •
Géza Fodor ==Recognition and honours==