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Paul Erdős (occasionally spelled Erdos or Erdös; Hungarian: Erdős Pál, pronounced [ˈɛrdøːʃ ˈpaːl]; 26 March 1913 – 20 September 1996) was an immensely prolific and notably eccentric Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory.


Paul Erdős was born in Budapest, Hungary on March 26, 1913.[1] He was the only surviving child of Anna and Lajos Erdős; his siblings died before he was born, aged 3 and 5. His parents were both Jewish mathematicians from a vibrant intellectual community.[2] His fascination with mathematics developed early—at the age of three, he could calculate how many seconds a person had lived.[3]

Both of Erdős's parents were high school mathematics teachers, and Erdős received much of his early education from them. Erdős always remembered his parents with great affection. At 16, his father introduced him to two of his lifetime favorite subjects—infinite series and set theory. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Erdős later published several articles in it about problems in elementary plane geometry.[4]

In 1934, at the age of 21, he was awarded a doctorate in mathematics.[5] Because anti-Semitism was increasing, he moved that same year to Manchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at Princeton University. At this time, he began to develop the habit of traveling from campus to campus. He would not stay long in one place and traveled back and forth among mathematical institutions until his death.

Another roof, another proof.
Paul Erdős[6]

Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life as a vagabond, traveling between scientific conferences and the homes of colleagues all over the world. He would typically show up at a colleague's doorstep and announce "my brain is open," staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom he (Erdős) should visit next. His working style has been humorously compared[by whom?] to traversing a linked list.

His colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities. (This quotation is often attributed incorrectly to Erdős himself.)[7] After 1971 he also took amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking the drug for a month.[8] Erdős won the bet, but complained during his abstinence that mathematics had been set back by a month: "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper." After he won the bet, he promptly resumed his amphetamine habit.

He had his own idiosyncratic vocabulary: he spoke of "The Book", an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF).[9] He accused the SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from The Book!". This later inspired a book entitled Proofs from THE BOOK.

Other idiosyncratic elements of Erdős' vocabulary include:

* Children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted by that Greek letter (ε));
* Women were "bosses";
* Men were "slaves";
* People who stopped doing math had "died";
* People who physically died had "left";
* Alcoholic drinks were "poison";
* Music was "noise";
* People who had married were "captured";
* People who had divorced were "liberated";
* To give a mathematical lecture was "to preach" and
* To give an oral exam to a student was "to torture" him/her.

Also, all countries which he thought failed to provide freedom to individuals as long as they did no harm to anyone else were classified as imperialist and given a name that began with a lowercase letter. For example, the U.S. was "samland" (after Uncle Sam), the Soviet Union was "joedom" (after Joseph Stalin), and Israel was "israel". For his epitaph he suggested, "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább").[10]

In 1952, during the McCarthy anti-communist investigations, the U.S. government denied Erdős, a Hungarian citizen, a re-entry visa into the United States, for reasons that have never been fully explained.[11] Teaching at Notre Dame at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the Immigration Service at periodic intervals. The government changed its mind in 1963 and Erdős resumed including American universities in his teaching and travels.
Erdős, Fan Chung, and her husband Ronald Graham, Japan 1986

Hungary, then a Communist nation, was under the hegemony of the Soviet Union. Although it curtailed the freedom of its citizens, in 1956 it gave Erdős the singular privilege of being allowed to enter and exit Hungary as he pleased. Erdős exiled himself voluntarily from Hungary in 1973 as a principled protest against his country's policy of denying entry to Israelis.[12]

During the last decades of his life, Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences and the UK Royal Society. Shortly before his death, he renounced his honorary degree from the University of Waterloo over what he considered to be unfair treatment of colleague Adrian Bondy.[13][14] He died "in action" of a heart attack on September 20, 1996, at the age of 83, while attending a conference in Warsaw, Poland. Erdős never married and had no children.

His life was documented in the film N Is a Number: A Portrait of Paul Erdős, made while he was still alive.

Mathematical work

Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler; Erdős published more papers, while Euler published more pages.[15] He wrote around 1,475 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity,[16] having 511 different collaborators in his lifetime.[17]

In terms of mathematical style, Erdős was much more of a "problem solver" than a "theory developer". (See "The Two Cultures of Mathematics"[18] 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."[19] Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did,[20] a pattern that extends to other prizes.[21] He did win the Wolf Prize, where his contribution is described as "for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over".[22] In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".

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 an elementary proof for the prime number theorem, along with Atle Selberg, which showed how combinatorics was an efficient method of counting collections. 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.[23]

Erdős problems

Throughout his career, Erdős would offer prizes for solutions to unresolved problems. These ranged from $25 for problems that he felt were just out of the reach of current mathematical thinking, to several thousand dollars for problems that were both difficult to attack and mathematically significant. There are thought to be at least a thousand such outstanding prizes, though there is no official or comprehensive list. The prizes remain active despite Erdős's death; Ronald Graham is the (informal) administrator of solutions. Winners can get either a check signed by Erdős (for framing only) or a cashable check from Graham.[24]

Perhaps the most notable of these problems is the Erdős conjecture on arithmetic progressions:

If the sum of the reciprocals of a sequence of integers diverges, then the sequence contains arithmetic progressions of arbitrary length.

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 problem is currently worth US$5000.[25]


His 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[26]

* Béla Bollobás (18 papers)
* Stefan Burr (27 papers)
* Fan Chung (14 papers)
* Zoltán Füredi (10 papers)
* Ron Graham (28 papers)
* András Gyárfás (15 papers)
* Richard R. Hall (14 papers)
* Istvan Joo (12 papers)
* Eric Milner (15 papers)
* Melvyn Nathanson (19 papers)
* Jean-Louis Nicolas (19 papers)
* János Pach (21 papers)
* George Piranian (14 papers)
* Carl Pomerance (23 papers)
* Richard Rado (18 papers)
* A. R. Reddy (11 papers)
* Alfréd Rényi (32 papers)
* Pal Revesz (10 papers)
* Vojtěch Rödl (11 papers)
* C. C. Rousseau (35 papers)
* Dick Schelp (42 papers)
* John Selfridge (14 papers)
* Miklós Simonovits (21 papers)
* Vera Sós (35 papers)
* Joel Spencer (23 papers)
* Ernst G. Straus (20 papers)
* Endre Szemerédi (29 papers)
* Paul Turán (30 papers)
* Zsolt Tuza (12 papers)

For other co-authors of Erdős, see the list of people with Erdős number 1 in List of people by Erdős number.

Erdős number
Main article: Erdős number

Because of his prolific output, friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number,[27] and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in light of the small world phenomenon).

It is said that Baseball Hall of Famer Hank Aaron has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day.[28] Erdős numbers have also been assigned to an infant, a horse, and several actors.[29]

The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1.[30] Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?"[31]

See also

* Cameron–Erdős conjecture
* Erdős Prize
* Erdős-Kac theorem
* List of things named after Paul Erdős
* Probabilistic method


1. ^ "Erdos biography". Retrieved 2010-05-29.
2. ^ The Budapest Jewish community of that day produced at least six remarkable thinkers besides Erdős: physicist and mathematician Eugene Wigner (Wigner Jenő in Hungarian), physicist Edward Teller (Teller Ede), physicist Leó Szilárd (Szilárd Leó), mathematician and polymath John von Neumann (Neumann János), physicist Dennis Gabor (Gábor Dénes), and philosopher Georg Lukács (Lukács György).
3. ^ Hoffman, p. 66.
4. ^ "?".
5. ^ Erdős's thesis advisor at the University of Budapest was Leopold Fejér (or Fejér Lipót), who was also the thesis advisor for John von Neumann, George Pólya and Paul (Pál) Turán.
6. ^ Cited in at least 20 books.
7. ^ Biography of Alfréd Rényi by J.J. O'Connor and E.F. Robertson
8. ^ Hill, J. Paul Erdos, Mathematical Genius, Human (In That Order)
9. ^ Schechter, Bruce (2000). My brain is open: The mathematical journeys of Paul Erdős. New York: Simon & Schuster. pp. 70–71. ISBN 0-684-85980-7.
10. ^ Hoffman, p. 3.
11. ^ "Erdos biography". School of Mathematics and Statistics, University of St Andrews, Scotland. January 2000. Retrieved 2008-11-11.
12. ^ László Babai and Joel Spencer. "Paul Erdős (1913–1996)" (PDF). Notices of the American Mathematical Society (American Mathematical Society) 45 (1).
13. ^ Letter from Erdős to University of Waterloo
14. ^ Transcription of October 2, 1996, article from University of Waterloo Gazette
15. ^ Hoffman, p. 42.
16. ^ Charles Krauthammer (September 27, 1996). "Paul Erdos, Sweet Genius". Washington Post: p. A25. "?".
17. ^ "The Erdős Number Project Data Files". 2009-05-29. Retrieved 2010-05-29.
18. ^ This essay is in Mathematics: Frontiers and Perspectives, Edited by V. I. Arnold, Michael Atiyah, Peter D. Lax and Barry Mazur, American Mathematical Society, 2000.
19. ^ Joel Spencer, "Prove and Conjecture!", a review of Mathematics: Frontiers and Perspectives. American Scientist, Volume 88, No. 6 November-December 2000
20. ^ Paths to Erdös — The Erdös Number Project
21. ^ 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 Lovasz received the Wolf prize, making 2 in all).
22. ^ "Wolf Foundation Mathematics Prize Page". Retrieved 2010-05-29.
23. ^ Melvin Henriksen. "Reminiscences of Paul Erdös (1913-1996)". Mathematical Association of America. Retrieved 2008-09-01.
24. ^ Charles Seife (5 April 2002). "Erdös's Hard-to-Win Prizes Still Draw Bounty Hunters". Science 296 (5565): 39. doi:10.1126/science.296.5565.39. PMID 11935003.
25. ^ p. 354, Soifer, Alexander (2008); The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators; New York: Springer. ISBN 978-0387746401
26. ^ List of collaborators of Erdős by number of joint papers, from the Erdős number project web site.
27. ^ "From Benford to Erdös". Radio Lab. 2009-09-30. No. 2009-10-09.
28. ^ Jerry Grossman. "Items of Interest Related to Erdös Numbers".
29. ^ Extended Erdős Number Project
30. ^ Michael Golomb's obituary of Paul Erdős
31. ^ Goffman, Casper (1969). "And what is your Erdős number?". American Mathematical Monthly 76: 791. doi:10.2307/2317868.


* Aigner, Martin; Günther Ziegler (2003). Proofs from THE BOOK. Berlin; New York: Springer. ISBN 3-540-40460-0.
* N Is a Number: A Portrait of Paul Erdős. Berlin; Heidelberg: Springer Verlag. 2005. ISBN 3-540-22469-6. - DVD
o 1993 documentary film "N Is a Number: A Portrait of Paul Erdős" by George Paul Csicsery.
* Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. London: Fourth Estate Ltd. ISBN 1-85702-811-2.
* Kolata, Gina (1996-09-24). "Paul Erdos, 83, a Wayfarer In Math's Vanguard, Is Dead". New York Times: pp. A1 and B8. Retrieved 2008-09-29.
* Bruce Schechter (1998). My Brain is Open: The Mathematical Journeys of Paul Erdős. Simon & Schuster. ISBN 0-684-84635-7.

External links

* Searchable collection of (almost) all papers of Erdős
* O'Connor, John J.; Robertson, Edmund F., "Erdös, Paul", MacTutor History of Mathematics archive, University of St Andrews, .
* Paul Erdős at the Mathematics Genealogy Project
* Jerry Grossman at Oakland University. The Erdös Number Project
* The Man Who Loved Only Numbers - Royal Society Public Lecture by Paul Hoffman (video)
* Erdős in Memphis 1996 part 2 at YouTube
* Erdős in Memphis 1996 part 3 at YouTube
* Radiolab: Numbers, with a story on Paul Erdős


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