Paul Erdős ( ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics.
He taught at various universities in the United States and Israel. Erdős's output was prolific; he published around 1,500 mathematical papers during his lifetime, many being collaborations with other mathematicians, making him arguably the most prolific mathematician in history. This prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.
He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle. He firmly believed mathematics to be a social activity, living a nomadic lifestyle with the sole purpose of writing mathematical papers with other mathematicians. He devoted his waking hours to mathematics, even into his later years; he died at a mathematics conference in Warsaw in 1996.
Early life and education Paul Erdős was born on 26 March 1913, in Budapest, Austria-Hungary, the only surviving child of Anna (née Wilhelm) and Lajos Erdős (né Engländer). His two sisters, aged three and five, both died of scarlet fever a few days before he was born. He was born to a well-to-do Hungarian-Jewish family, both of his parents worked as high school mathematics teachers. His fascination with mathematics developed early. He was raised partly by a German governess because his father was held captive in Siberia as an Austro-Hungarian prisoner of war during 1914–1920, causing his mother to have to work long hours to support their household. His father had taught himself English while in captivity but mispronounced many words. When Lajos later taught his son to speak English, Paul learned his father's pronunciation, which he continued to use for the rest of his life.
He taught himself to read through mathematics texts that his parents left around in their home. By the age of five, given a person's age, he could calculate in his head how many seconds they had lived. Due to his sisters' deaths, he had a close relationship with his mother, with the two of them reportedly sharing the same bed until he left for college.
When he was 16, his father introduced him to two subjects that would become lifetime favourites—infinite series and set theory. In high school, Erdős became an ardent solver of the problems that appeared each month in KöMaL, the "Mathematical and Physical Journal for Secondary Schools".
Erdős began studying at the University of Budapest when he was 17 after winning a national examination. At the time, admission of Jews to Hungarian universities was severely restricted under the numerus clausus.
During 1933, Erdős and several other students, including George Szekeres, Esther Klein (later Szekeres), her lifelong friend Márta Wachsberger (later Svéd), and George Svéd, met frequently, often at the Anonymous statue in City Park, to discuss mathematics. Klein proposed a problem, offering her proof, about which Szekeres and Erdős wrote a paper that generalised the result in 1935. Erdős dubbed the original problem the "Happy ending problem" because it resulted in the marriage of George and Esther Szekeres.
By the time he was 20, Erdős had found a proof for Bertrand's postulate. In 1934, at the age of 21, he was awarded a doctorate in mathematics. Erdős's thesis advisor was Lipót Fejér, who was also the thesis advisor for John von Neumann, George Pólya, and Pál Turán.
Career 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.
Because he was Jewish, Hungary was dangerous, so he left the country, relocating to the United States in 1938. In 1938, he accepted his first American position as a scholarship holder at the Institute for Advanced Study, Princeton, New Jersey, for the next ten years. Despite outstanding papers with Mark Kac and Aurel Wintner on probabilistic number theory, Pál Turán in approximation theory, and Witold Hurewicz on dimension theory, his fellowship was not continued, and Erdős was forced to take positions as a wandering scholar at University of Pennsylvania, Notre Dame, Purdue, Stanford, and Syracuse. He would not stay long in one place, instead traveling among mathematical institutions until his death.
Described by his biographer, Paul Hoffman, as "probably the most eccentric mathematician in the world", Erdős spent most of his adult life living out of a suitcase. 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. During his visits, Erdős expected his hosts to lodge him, feed him, and do his laundry, along with anything else he needed, as well as arrange for him to get to his next destination.
In 1941, Erdős and two others would be arrested by authorities for looking around a secret radio tower, with Erdős reportedly telling police that rather than noticing the signs against trespassing, "I was thinking about mathematical theorems." The FBI would maintain surveillance and a file on Erdős into the 1970s, but never found anything incriminating.
In 1943, Erdős began a part-time appointment at Purdue University. Creative Commons Attribution-ShareAlike 4.0 International License. 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.
Near the end of 1948 Erdős was able to return to Hungary for a visit, where he was reunited with his surviving family and friends. For the next three years he traveled frequently between England and the United States, before accepting a temporary post at the University of Notre Dame, Indiana, in 1952. The post gave him complete freedom to travel to do joint research whenever he wanted, but, despite encouragement from the university and his friends, he would not accept the offer on a permanent basis. Hungary at the time was under the Warsaw Pact with the Soviet Union. Although Hungary limited the freedom of its own citizens to enter and exit the country, in 1956 it gave Erdős the exclusive privilege of being allowed to enter and exit the country as he pleased.
As a result of the Red Scare and McCarthyism, 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. He requested reconsideration from the U.S. Immigration Services at periodic intervals during the early 1960s, and visa was finally granted in November 1963. During this period he spent around 10 years in Israel. He was given a position for three months at the Hebrew University of Jerusalem, and then a "permanent visiting professor" position at the Technion in Haifa.
In 1963, the United States Immigration and Naturalization Service granted Erdős a visa, and he resumed teaching at and traveling to American institutions. Ten years later, in 1973, the 60-year-old Erdős voluntarily left Hungary.
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 2023According to Facts about Erdös Numbers and the Collaboration Graph, using the Mathematical Reviews data base, the next highest article count is roughly 823. – mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.
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." This article is a review of Mathematics: Frontiers and Perspectives 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.From "trails to Erdos" , by DeCastro and Grossman, in The Mathematical Intelligencer, vol. 21, no. 3 (Summer 1999), 51–63: A careful reading of Table 3 shows that although Erdos never wrote jointly with any of the 42 [Fields] medalists (a fact perhaps worthy of further contemplation)... there are many other important international awards for mathematicians. Perhaps the three most renowned...are the Rolf Nevanlinna Prize, the Wolf Prize in Mathematics, and the Leroy P. Steele Prizes. ... Again, one may wonder why KAPLANSKY is the only recipient of any of these prizes who collaborated with Paul Erdös. (After this paper was written, collaborator Lovász received the Wolf prize, making 2 in all). 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