# Paul Erdős

Paul Erdős at a seminar in Budapest (autumn 1992)

Paul Erdős [ ˈɛrdøːʃ ] ( Hungarian Erdős Pál ; born March 26, 1913 in Budapest , Austria-Hungary , † September 20, 1996 in Warsaw , Poland ) was one of the most important mathematicians of the 20th century. Paul Erdős worked with hundreds of colleagues in the fields of combinatorics , graph theory and number theory . Erdős is known in connection with the Erdős number . He also came up with the idea for the BOOK in which God keeps the perfect proof of mathematical theorems. An attempt to come close to this book is The Book of Evidence by Aigner and Ziegler , published in 1998 and on which he was still involved before his death.

## Life

Paul Erdős was born the third child into a Jewish family. After his two sisters had died at the age of three and five - even before he was born - he was the only child of Anna and Lajos Erdős. His parents were both math teachers and religious freethinkers, which was carried over to Paul Erdős. His father was captured in the first year of the war in 1914 as a member of the Austro-Hungarian army in an attack by the Russians in Galicia and spent several years as a prisoner of war in Siberia . While his mother was teaching, Paul was raised by a German governess. By the time he was three years old, he could count, and by the time he was four he could calculate in his head how many seconds they had lived with family friends. His mother, for fear of contagious diseases that caused both of his sisters to die, did not let him go to public school, but instead gave him a private tutor. As a child, Erdős was very dependent on his mother, who died in 1971, in everyday matters. B. according to own statements, only to tie the shoelaces at the age of eleven. Even when he was supposed to go to high school, he only went to school every other year as his mother often changed her mind. His mother became director of the school under the brief rule of communist Béla Kun (1919), but was dismissed under the rule of Admiral Miklós Horthy , which began in 1920 . The anti-Semitism fostered by this government made many Jewish scientists (including Edward Teller , John von Neumann , Leó Szilárd and Eugene Wigner ) leave the country. In 1920 his father returned from captivity in Siberia. He had taught himself English while a prisoner of war, but without mastering the pronunciation, and transferred this accent to his son. At the age of 17 (1930) Paul Erdős enrolled at the university. He was only able to do this because the admission restrictions of 1920 were relaxed again in 1928, and Jews were able to study again as winners of national competitions. Only four years later (1934) he obtained a doctorate in mathematics. Since anti-Semitism increased more and more, he went to Harold Davenport in Manchester on a scholarship in the same year , but traveled widely within England and met u. a. Hardy in Cambridge and Stanisław Ulam, who also emigrated .

In 1938 he took his first position in the USA, as a scholarship holder, in Princeton ( New Jersey ). He did not keep this for long, however, as the Princeton management team considered him “peculiar and unconventional”, and he accepted an invitation from Ulam to Madison. It was around this time that he began to develop the habit of traveling from campus to campus. He never stayed in one place for long and traveled back and forth between mathematical institutes until his death.

In 1941 Paul Erdős went on a trip with his colleagues Arthur Stone and Shizuo Kakutani . They wanted to look out to sea from a raised tower. Just thinking about math, they overlooked a sign saying “No entry”. They took a few souvenir photos and were later arrested and interrogated by the FBI for espionage . The misunderstanding soon cleared up, but his entry on an FBI file hurt him later in the McCarthy era .

Only after the war did he find out about the fate of his relatives in Hungary, many of whom had perished in the Holocaust. He was very worried about his mother, who had survived the Holocaust. His father died of a heart attack in 1942. When he visited his mother and friends ( Paul Turan , Vera T. Sós , Miklós Simonovits and others) in Hungary in December 1948 after a ten-year break , he was only able to leave Hungary again in February 1949, as Stalin crossed the borders in the beginning of the Cold War had locked down. He then commuted back and forth between England and the USA for three years before accepting a position at the US University of Notre Dame in 1952 .

When he wanted to travel to a conference in Amsterdam in 1954 , after an investigation before a McCarthy commission he was told that if he left the USA he would not be allowed to re-enter, which did not prevent Erdős from going to the conference.

Since the Netherlands and England also imposed travel and residence restrictions on him, he accepted a position at the Hebrew University of Jerusalem in the 1960s . Despite many attempts, he was not allowed to enter the USA again until 1963. No reason was given officially, but the files indicate that his arrest in 1941 and his contacts with the Chinese number theorist Loo-Keng Hua were the cause.

Erdős kept his position in Jerusalem formally for 30 years: however, he always traveled from university to university to work with other mathematicians. He published around 1,500 joint articles, more than any other mathematician. This also gave rise to the half-joking Erdős number . The 509 mathematicians who worked directly with him have the Erdős number 1; those who did not work with Erdős but with someone with Erdős number 1 have Erdős number 2; etc.

He slept only four to five hours a day and got excited with coffee, caffeine tablets and amphetamines , which he was prescribed because of depression after his mother's death. In 1979, his friend Ronald Graham offered him a \$ 500 bet because he was concerned that Erdős was addicted: he would not be able to last 30 days without a stimulant. He held out the 30 days, but said the bet had set math back by a month because he couldn't put a thought on paper. After the bet, he resumed amphetamine use.

Paul Erdős led a materially simple life devoted to mathematics. With the prize money he won, he supported gifted students, donated them or used them as prize money for difficult tasks. In 1977 he donated the Israeli Erdős Prize , named in honor of his parents. The Paul Erdős Prize of the Hungarian Academy of Sciences is named in his honor.

In 1974 Erdős was elected to the American Academy of Arts and Sciences , in 1980 to the National Academy of Sciences . In 1983 he received the Wolf Prize . In 1983 he gave a plenary lecture at the ICM in Warsaw ( Extremal problems in number theory, combinatorics and geometry ) and in 1970 he was invited speaker at the ICM in Nice ( On the application of combinatorial analysis to number theory, geometry and analysis ).

In September 1996 Erdős took part in a conference in Warsaw on graph theory. There he died on September 20th as a result of two heart attacks. Erdős, who suffered from cardiac arrhythmias for the last decade of his life, is buried in the Jewish cemetery in Rákoskeresztúr (now part of Budapest), where his parents are also lying.

## plant

Erdős' main areas of work were number theory and combinatorics . He was also a pioneer in the application of probabilistic arguments in number theory and graph theory. Erdős was not so interested in building theories as he was in solving specific problems with the simplest, elegant and “insightful” evidence possible.

In 1931, while still a student in Budapest, he found an elegant elementary proof of Bertrand's conjecture that there would always be a prime between and (the proof was already provided by Pafnuti Lvowitsch Chebyshev ). In 1948 he attracted attention when, together with Atle Selberg, he gave an "elementary" proof of the prime number theorem (i.e. without function theory) , after Chebyshev had already given estimates using "elementary" methods in the 19th century. The question of what part of Erdős had in the evidence has long been controversial. According to the “eyewitness” Ernst Gabor Straus , Erdős' contribution (which arose in a seminar by Paul Turán at Princeton) was also of essential importance for Selberg's evidence. The starting point was an inequality proved by Selberg , which Turan used for an elementary proof of Dirichlet's theorem about prime numbers in arithmetic sequences . Erdős proved with the inequality a generalization of Chebyshev's theorem mentioned above (namely that there is a prime number between and , fixed and sufficiently large). When he communicated this to Selberg, Selberg did not want to believe it at first, as it would provide an elementary proof of the prime number theorem, which he had previously tried in vain. Selberg declined a joint publication offered by Erdős and eventually published evidence that circumvented Erdős' contribution. Among other things, Selberg received the Fields Medal for this, Erdős came away empty-handed. According to the memoirs of Ernst Straus, Hermann Weyl played an important role in this, who protected Selberg, who was closer to his mathematical approach, and ensured that the Annals of Mathematics rejected Erdős' article. Erdős published work with Mark Kac on the probabilistic interpretation of the prime number theorem and in 1939 proved Erdős-Kac's theorem , which - roughly speaking - says that the number of prime factors of a natural number is "normally distributed". Erdős heard Kac say the sentence as a conjecture in a Princeton lecture and came up with a proof shortly after the lecture ended. ${\ displaystyle n> 1}$${\ displaystyle n}$${\ displaystyle 2n}$${\ displaystyle \, x}$${\ displaystyle \, x (1+ \ epsilon)}$${\ displaystyle \ epsilon}$${\ displaystyle x}$

Erdős also wrote an influential work on prime number twins from 1940. In 1946, together with Arthur Herbert Copeland , he proved that the Copeland-Erdős number named after them is a normal number . In 1975 he proved with John Selfridge the set of Erdős-Selfridge , that the product of successive natural numbers can not be a real potency.

In combinatorics he worked in the theory of extremal graphs, combinatorial questions of elementary geometry and in Ramsey theory , which predicts the appearance of orders in sufficiently large random structures. Here he was involved in the Erdős-Szekeres theorem of 1935, which quantitatively provides much more precise information in the Ramsey theory. He also introduced the idea of ​​asymptotic estimates from number theory to combinatorics. This area is sometimes called combinatorial number theory .

Examples of his results in combinatorics are a generalization of the "Happy Ending Theorem" with George Szekeres in 1935 (a sufficiently large number of points in the plane in a general position - ie: no three points lie on a straight line - contains any given number of Points that form a convex polygon). In this work, Szekeres (then a chemical engineer student) rediscovered theorems of Ramsey , which were soon followed by Erdős et al. a. were expanded to the Ramsey theory. In 1957 he proved the theorem that for everyone there is always a graph with the “chromatic number” (minimum number of colors necessary to differently color adjacent corners) in which all cycles (closed paths) are longer than . ${\ displaystyle k, m}$${\ displaystyle k}$${\ displaystyle m}$

In a series of works with Alfréd Rényi from 1959 to 1968, he developed the theory of random graphs with corners and edges. In particular, they were able to prove phase transitions for the appearance of new properties and structures depending on the size of the graph . This work also had an impact on computer science. ${\ displaystyle m}$${\ displaystyle n}$${\ displaystyle (m, n)}$

As well as his sentences, he is known for his guesswork . One of these conjectures is that in every set of natural numbers for which the sum of the reciprocal values ​​of the elements diverges, there are arithmetic sequences of any length (assumed in a work with Turan 1936). For the proof of a slightly weaker version, the mathematician Endre Szemerédi received 1,000 dollars from Erdős. Hillel Fürstenberg later gave a proof of ergodic theory .

Erdős played a leading role in the development of infinite combinatorics in set theory. Together with András Hajnal , Richard Rado and others, he investigated partition properties of ordinal numbers and uncountable cardinal numbers and proved variants and generalizations of Ramsey's theorem, see Erdős-Rado's theorem .

Erdős also achieved important results in numerical mathematics, especially in the theory of the approximation of functions, e.g. B. in a work with Paul Turan 1937, in which they showed that the Lagrangian interpolation polynomials of any continuous function converge on average to this function for any weight functions at the support points formed from the roots of a system of orthogonal polynomials .

## Literature and Sources

• M. Aigner, G. Ziegler: The BOOK of evidence . Heidelberg, Springer 2003, ISBN 3-540-40185-7 .
An attempt to make Erdős' idea of ​​God's “BOOK” of elegant evidence a reality, including some from Erdős himself.
• GP Csicsery: N is a Number. A Portrait of Paul Erdős. The Story of a Wandering Mathematician obsessed with unsolved Problems. (a video). Heidelberg, Springer (Springer VideoMATH) 2000, ISBN 3-540-92642-9 .
A video about Erdős man and his work. Contains some computer animations that illustrate his research.
• G. Halasz, L. Lovasz, M. Simonovits, V. Sos (Eds.): Paul Erdős and His Mathematics. 2 vols. Heidelberg, Springer 2002, ISBN 3-540-42236-6 .
Erdős' most important original works summarized in two volumes.
• Bruce Schechter: My mind is open. The mathematical journeys of Paul Erdős . Basel, Birkhäuser 1999, ISBN 3-7643-6083-6 .
Is considered more objective than Hoffman's biography. (The German edition is only available as an antiquarian, the English has ISBN 0-684-85980-7 )
• Paul Hoffman: The man who loved numbers . Ullstein 1998, ISBN 3-550-06978-2 .
• Vera T. Sós: Paul Erdős, 1913–1996. In: Aequationes mathematicae 54 (1997) , pp. 205-220.