View source for Biography:Paul Erdős

{{short description|Hungarian mathematician (1913–1996)}}
{{Eastern name order|Erdős Pál}}
{{Infobox scientist
|name = Paul Erdős
|image = Erdos budapest fall 1992 (cropped).jpg
|caption = Paul Erdős in 1992
|alt = Erdős looking toward the camera
|birth_place = [[Place:Budapest|Budapest]], [[Place:Austria-Hungary|Austria-Hungary]]
|death_date = {{death date and age|1996|9|20|1913|3|26|df=y}}
|death_place = [[Place:Warsaw|Warsaw]], Poland
|citizenship = 
|nationality = Hungarian
|fields = [[Mathematics]]
|workplaces = {{plainlist|Victoria University of Manchester
* [[Organization:Princeton University|Princeton University]]
* [[Organization:Purdue University|Purdue University]]
* [[Organization:University of Pennsylvania|University of Pennsylvania]]
* University of Notre Dame
* [[Organization:Stanford University|Stanford University]]
* [[Organization:Syracuse University|Syracuse University]]
* [[Organization:Hebrew University of Jerusalem|Hebrew University of Jerusalem]]
* Technion – Israel Institute of Technology}}
|alma_mater = Royal Hungarian Pázmány Péter University
|doctoral_advisor = [[Biography:Lipót Fejér|Lipót Fejér]]
|academic_advisors = 
|doctoral_students = {{plainlist|[[Biography:Joseph Kruskal|Joseph Kruskal]]
*[[Biography:George B. Purdy|George B. Purdy]]
*Alexander Soifer
*[[Biography:Béla Bollobás|Béla Bollobás]]<ref>{{cite web|url=http://genealogy.math.ndsu.nodak.edu/id.php?id=19470|title=Mathematics Genealogy Project|access-date=13 August 2012}}</ref>}}
|notable_students = 
|known_for = Namesakes
A very large number of results and conjectures (more than 1,500 articles), and a very large number of coauthors (more than 500)
|author_abbrev_bot = 
|author_abbrev_zoo = 
|influences = 
|influenced = 
|awards = Wolf Prize (1983/84)<br>AMS Cole Prize (1951)
|signature = <!--(filename only)-->
}}

'''Paul Erdős''' ({{lang-hu|Erdős Pál}} {{IPA-hu|ˈɛrdøːʃ ˈpaːl|}}; 26 March 1913 – 20 September 1996) was a Hungarian [[Biography:Mathematician|mathematician]]. He was one of the most prolific mathematicians and producers of mathematical [[Conjecture|conjecture]]s<ref>{{Cite web|url=https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/|title=The Sum-Product Problem Shows How Addition and Multiplication Constrain Each Other|website=Quanta Magazine|date=6 February 2019|access-date=October 6, 2019}}</ref> of the 20th century.<ref>{{cite web |title=Paul Erdős |url=https://www.britannica.com/EBchecked/topic/191138/Paul-Erdos |publisher="Encyclopædia Britannica |first=Paul |last=Hoffman |date=July 8, 2013}}</ref> {{nobr|Erdős}} pursued and proposed problems in [[Discrete mathematics|discrete mathematics]], [[Graph theory|graph theory]], [[Number theory|number theory]], [[Mathematical analysis|mathematical analysis]], [[Approximation theory|approximation theory]], [[Set theory|set theory]], and [[Probability theory|probability theory]].<ref name="britannica">{{cite web|url=https://www.britannica.com/EBchecked/topic/191138/Paul-Erdos|title=Paul Erdos - Hungarian mathematician|website=Britannica.com|access-date=2017-12-02}}</ref> Much of his work centered around 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 [[Unsolved:Open problem|open problem]]s, rather than developing or exploring new areas of mathematics.

Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed.<ref>According to {{cite web |title=Facts about Erdös Numbers and the Collaboration Graph |url=http://oakland.edu/enp/trivia/}}, using the Mathematical Reviews data base, the next highest article count is roughly 823.</ref> He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathematicians. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddball".<ref>{{cite magazine |last=Lemonick |first=Michael D. |date=March 29, 1999 |title=Paul Erdos: The Oddball's Oddball |url=http://www.time.com/time/magazine/article/0,9171,990598,00.html |url-status=dead |magazine=Time |archive-url=https://web.archive.org/web/20120106094724/https://time.com/time/magazine/article/0,9171,990598,00.html |archive-date=January 6, 2012}}</ref> He devoted his waking hours to mathematics, even into his later years—indeed, his death came at a mathematics conference in [[Place:Warsaw|Warsaw]].<ref name="NYT_obit"/> Erdős's prolific output with co-authors 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.

==Life==
Paul Erdős was born on 26 March 1913, in [[Place:Budapest|Budapest]], [[Place:Austria-Hungary|Austria-Hungary]],<ref>{{cite web |url=http://www.gap-system.org/~history/Biographies/Erdos.html |title=Erdos biography |publisher=Gap-system.org |access-date=2010-05-29 |url-status=dead |archive-url=https://web.archive.org/web/20110607183318/http://www.gap-system.org/~history/Biographies/Erdos.html |archive-date=2011-06-07 }}</ref> the only surviving child of Anna (née Wilhelm) and Lajos Erdős (né Engländer).<ref name="BakerBollobas">{{Cite journal | last1 = Baker | first1 = A. | last2 = Bollobas | first2 = B. |  doi = 10.1098/rsbm.1999.0011 | title = Paul Erdős 26 March 1913 – 20 September 1996: Elected For.Mem.R.S. 1989 | journal = Biographical Memoirs of Fellows of the Royal Society | volume = 45 | pages = 147–164 | year = 1999 | doi-access = free }}</ref><ref name="ChernHirzebruch2000-p294" /> His two sisters, aged three and five, both died of [[Medicine:Scarlet fever|scarlet fever]] a few days before he was born.<ref>{{cite web|url=http://www-history.mcs.st-and.ac.uk/history/Biographies/Erdos.html |title=Paul Erdős |access-date=2015-06-11}}</ref> His parents, both Jewish, were high school mathematics teachers. His fascination with mathematics developed early—he was often left home by himself because his father was held captive in Siberia as an Austro-Hungarian [[Social:Prisoner of war|prisoner of war]] during 1914–1920,<ref name="ChernHirzebruch2000-p294">{{cite book|last1=Chern|first1=Shiing-Shen|last2=Hirzebruch|first2=Friedrich|title=Wolf Prize in Mathematics|url=https://books.google.com/books?id=cSUSUfvT3PoC&pg=PA294|year=2000|publisher=World Scientific|isbn=978-981-02-3945-9|page=294}}</ref> 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 acquired his unique pronunciations, which he continued to use for the rest of his life.<ref name=":4">{{Cite web |title=Paul Erdős - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Erdos/ |access-date=2022-07-06 |website=Maths History |language=en}}</ref>

He taught himself to read through mathematics texts that his parents left around in their home. By the age of four, given a person's age, he could calculate in his head how many seconds they had lived.{{sfn|Hoffman|1998|p=[https://archive.org/details/manwholovedonlyn00hoff/page/n77/mode/2up 66]}} 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.<ref>{{cite web|last1=Hoffman|first1=Paul|title="Paul Erdős: The Man Who Loved Only Numbers" video lecture|url=https://www.youtube.com/watch?v=9634A0iBw7w|website=YouTube|publisher=The University of Manchester|access-date=2017-03-17|date=1 July 2016}}</ref><ref>{{cite news |last1=Alexander |first1=James |title=Planning an Infinite Stay |url=https://archive.nytimes.com/www.nytimes.com/books/98/09/27/reviews/980927.27alexant.html |access-date=6 May 2022 |work={{wipe|The New York Times}} |date=September 27, 1998}}</ref>

When he was 16, his father introduced him to two subjects that would become lifetime favourites—[[Series (mathematics)|infinite series]] and [[Set theory|set theory]]. In high school, Erdős became an ardent solver of the problems that appeared each month in ''[[Középiskolai Matematikai és Fizikai Lapok|KöMaL]]'', the "Mathematical and Physical Journal for Secondary Schools".<ref>{{cite web |url=http://www.cs.uchicago.edu/files/tr_authentic/TR-2001-11.ps |title=Paul Erdős just left town |first=László |last=Babai |url-status=dead |archive-url=https://web.archive.org/web/20110609174744/http://www.cs.uchicago.edu/files/tr_authentic/TR-2001-11.ps |archive-date=2011-06-09 }}</ref>

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''.<ref name=":4" /><ref name=":0">{{harvnb|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/120 120]}}</ref> By the time he was 20, he had found a [[Mathematical proof|proof]] for [[Bertrand's postulate|Chebyshev's theorem]].<ref name=":0"/> In 1934, at the age of 21, he was awarded a doctorate in mathematics.<ref name=":0" /> Erdős's thesis advisor was [[Biography:Lipót Fejér|Lipót Fejér]], who was also the thesis advisor for [[Biography:John von Neumann|John von Neumann]], [[Biography:George Pólya|George Pólya]], and [[Biography:Pál Turán|Paul (Pál) Turán]]. He took up a post-doctoral fellowship at Manchester, as Jews in Hungary were suffering oppression under the [[History:Kingdom of Hungary (1920–1946)|authoritarian regime]]. While there he met [[Biography:G. H. Hardy|Godfrey Harold Hardy]] and [[Biography:Stanislaw Ulam|Stan Ulam]].<ref name=":4" />

Because he was Jewish, Erdős decided Hungary was dangerous and left the country, relocating to the United States in 1938.<ref name=":0" /> Many members of Erdős's family, including two of his aunts, two of his uncles, and his father, died in Budapest during World War II. His mother was the only one that survived. He was living in America and working at the Princeton Institute for Advanced Study at the time.<ref name=":0" /><ref name="documentary">{{cite video | author=Csicsery, George Paul | title=N Is a Number: A Portrait of Paul Erdős | publisher=Springer Verlag | location=Berlin; Heidelberg |date=2005 | isbn = 3-540-22469-6}}</ref> However, his fellowship at Princeton only got extended by 6 months rather than the expected year due to Erdős not conforming to the standards of the place; they found him "uncouth and unconventional".<ref name=":4" />

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.<ref name=":1">{{harvnb|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/121 121]}}</ref> 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.<ref name=":1" /> 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.<ref name=":1" />

Ulam left his post at the [[Organization:University of Wisconsin–Madison|University of Wisconsin–Madison]] in 1943 to work on the [[Physics:Manhattan Project|Manhattan Project]] in Los Alamos, New Mexico with other mathematicians and physicists. He invited Erdős to join the project, but the invitation was withdrawn when Erdős expressed a desire to return to Hungary after the war.<ref name=":4" />

On 20 September 1996, at the age of 83, he had a [[Medicine:Myocardial infarction|heart attack]] and died while attending a conference in [[Place:Warsaw|Warsaw]].<ref name=":2">{{harvnb|Bruno|2003|p=[https://archive.org/details/mathmathematicia00brun/page/122 122]}}</ref> These circumstances were close to the way he wanted to die. He once said,

{{Blockquote|text=I want to be giving a lecture, finishing up an important proof on the blackboard, when someone in the audience shouts out, 'What about the general case?'. I'll turn to the audience and smile, 'I'll leave that to the next generation,' and then I'll keel over.<ref name=":2" />}}

Erdős never married and had no children.<ref name="BakerBollobas" /> He is buried next to his mother and father in grave 17A-6-29 in the Jewish Kozma Street Cemetery in Budapest.<ref name="agt">{{cite web|url=http://www.agt.bme.hu/varga/foto/izraelita/erdos-p.html|website=agt.bme.hu|title=Erdős Pál sírja - grave 17A-6-29|access-date=2017-12-02|archive-url=https://web.archive.org/web/20160404120811/http://www.agt.bme.hu/varga/foto/izraelita/erdos-p.html|archive-date=2016-04-04|url-status=dead}}</ref> For his epitaph, he suggested "I've finally stopped getting dumber." (Hungarian: ''"Végre nem butulok tovább"'').{{sfn|Hoffman|1998|p=[https://archive.org/details/manwholovedonlyn00hoff/page/n15/mode/2up 3]}}

Erdős's name contains the Hungarian letter "ő" ("o" with double acute accent), but is often incorrectly written as ''Erdos'' or ''Erdös'' either "by mistake or out of typographical necessity".<ref>The full quote is "Note the pair of long accents on the "ő," often (even in Erdos's own papers) by mistake or out of typographical necessity replaced by "ö," the more familiar German umlaut which also exists in Hungarian.", from {{cite book |title=Combinatorics, Paul Erdős is eighty |first1=Paul |last1=Erdős |first2=D. |last2=Miklós |first3=Vera T. |last3=Sós |year=1996}}</ref>

==Career==
In 1934, Erdős moved to Manchester, England, to be a guest lecturer. 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 [[Biography:Mark Kac|Mark Kac]] and [[Biography:Aurel Wintner|Aurel Wintner]] on [[Probabilistic number theory|probabilistic number theory]], [[Biography:Pál Turán|Pál Turán]] in approximation theory, and Witold Hurewicz on dimension theory, his fellowship was not continued, and
Erdos was forced to take positions as a wandering scholar at the [[Organization:University of Pennsylvania|UPenn]], Notre Dame, [[Organization:Purdue University|Purdue]], Stanford, and [[Organization:Syracuse University|Syracuse]].{{sfn|Bollobás|1996|pp=4}} He would not stay long in one place, instead traveling among mathematical institutions until his death.

As a result of the Red Scare and McCarthyism,<ref>{{Cite web |date=2023-07-28 |title=The wandering mathematician: Paul Erdos |url=https://www.thearticle.com/the-wandering-mathematician-paul-erdos |access-date=2023-09-09 |website=TheArticle}}</ref><ref>{{Cite web |title=Paul Erdős - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Erdos/ |access-date=2023-09-09 |website=Maths History |language=en}}</ref><ref>{{Cite web |last=Sack |first=Harald |date=2018-09-20 |title=What's your Erdös Number? – The bustling Life of Mathematician Paul Erdös {{!}} SciHi Blog |url=http://scihi.org/bustling-life-paul-erdoes/ |access-date=2023-09-10 |language=en-US}}</ref> in 1954, the Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States.<ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/history/Biographies/Erdos.html |title=Erdos biography |access-date=2008-11-11 |publisher=School of Mathematics and Statistics, University of St Andrews, Scotland|date=January 2000}}</ref> Teaching at the University of 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 U.S. Immigration Services at periodic intervals. At some point he moved to live in Israel, and was given a position for three months at the Hebrew University in Jerusalem, and then a "permanent visiting professor" position at the Technion. 

[[File:ronald graham couple with erdos 1986.jpg|right|thumb|Counter-clockwise from left: Erdős, Fan Chung, and her husband Ronald Graham, Japan 1986]]

Hungary at the time was under the [[Organization:Warsaw Pact|Warsaw Pact]] with the [[Place:Soviet Union|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. 

In 1963, the U.S. Immigration 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.<ref>{{cite journal|title=Paul Erdős (1913–1996) |journal=[[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]]|publisher=American Mathematical Society|url=https://www.ams.org/notices/199801/comm-erdos.pdf|volume=45|issue=1|first1=László  |last1=Babai  |first2=Joel|last2=Spencer}}</ref>

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.<ref name="Erdõs Fellow RS">{{cite journal | last1=Baker | first1=A. | last2=Bollobás | first2=B. | title=Paul Erdõs. 26 March 1913 — 20 September 1996 | journal=Biographical Memoirs of Fellows of the Royal Society | publisher=The Royal Society | volume=45 | year=1999 | issn=0080-4606 | doi=10.1098/rsbm.1999.0011 | pages=147–164| s2cid=123517792 }}</ref> He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1977.<ref>{{cite web|url=https://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00000161 |archive-url=https://web.archive.org/web/20200728075004/https://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00000161 |title=P. Erdös (1913 - 1996) |publisher=Royal Netherlands Academy of Arts and Sciences |archive-date=28 July 2020}}</ref> 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.<ref>{{cite web|last1=Erdős|first1=Paul|title=Dear President Downey|url=http://ecp6.jussieu.fr/pageperso/bondy/uw/letters/erdos.pdf|access-date=2014-07-08|archive-url=https://web.archive.org/web/20051015141515/http://ecp6.jussieu.fr/pageperso/bondy/uw/letters/erdos.pdf|archive-date=2005-10-15|date=4 June 1996|quote=With a heavy heart I feel that I have to sever my connections with the University of Waterloo, including resigning my honorary degree which I received from the University in 1981 (which caused me great pleasure). I was very upset by the treatment of Professor Adrian Bondy. I do not maintain that Professor Bondy was innocent, but in view of his accomplishments and distinguished services to the University I feel that 'justice should be tempered with mercy.'}}</ref><ref>[http://communications.uwaterloo.ca/Gazette/1996/October02/Arbitrator%20upholds%20math%20prof's%20dismissal Transcription of October 2, 1996, article] from University of Waterloo Gazette ([https://web.archive.org/web/20120427141236/http://www.communications.uwaterloo.ca/Gazette/1996/October02/Arbitrator%20upholds%20math%20prof%27s%20dismissal archive]) {{webarchive |url=https://web.archive.org/web/20101123212756/http://communications.uwaterloo.ca/Gazette/1996/October02/Arbitrator%20upholds%20math%20prof's%20dismissal |date=November 23, 2010 }}</ref>

===Mathematical work===
Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with [[Biography:Leonhard Euler|Leonhard Euler]]; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by himself.{{sfn|Hoffman|1998|p=[https://archive.org/details/manwholovedonlyn00hoff/page/n55/mode/2up 42]}} Erdős wrote around 1,525 mathematical articles in his lifetime,<ref>{{cite web |url=http://www.oakland.edu/enp/pubinfo/ |title=Publications of Paul Erdös |first=Jerry |last=Grossman |access-date=2011-02-01}}</ref> mostly with co-authors. He strongly believed in and practiced mathematics as a social activity,<ref>{{Cite news | newspaper={{wipe|The Washington Post}} | first=Charles | last=Krauthammer | title=Paul Erdos, Sweet Genius | date=September 27, 1996| page=A25 | url=http://www.fmf.uni-lj.si/~mohar/Erdos.html| author-link=Charles Krauthammer }} {{cite web |url=http://theory.cs.uchicago.edu/erdos/wash-post2.ps |title= ? }}</ref> having 511 different collaborators in his lifetime.<ref>{{cite web|url=http://www.oakland.edu/enp/thedata/ |title=The Erdős Number Project Data Files |publisher=Oakland.edu |date=2009-05-29 |access-date=2010-05-29}}</ref>

In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics"<ref>{{cite book |chapter=The Two Cultures of Mathematics |first= Timothy |last= Gowers |title= Mathematics: Frontiers and Perspectives |editor-first1= V. I. |editor-last1=Arnold |editor-first2=Michael |editor-last2=Atiyah |editor-first3=Peter D. |editor-last3=Lax |editor-first4=Barry |editor-last4=Mazur |publisher=American Mathematical Society |year=2000 |chapter-url=http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf |isbn=978-0821826973}}</ref> by [[Biography:Timothy Gowers|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 [[Theorem|theorem]]s and conjectures throughout his illustrious career."<ref>{{cite journal |first=Joel |last=Spencer |title=Prove and Conjecture! |journal=American Scientist |volume=88 |issue=6 |date=November–December 2000}} This article is a review of ''Mathematics: Frontiers and Perspectives''</ref> Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did,<ref name="oakland">{{cite web|url=http://www.oakland.edu/enp/erdpaths/|title=Paths to Erdős - The Erdős Number Project- Oakland University|website=oakland.edu|access-date=2017-12-02}}</ref> a pattern that extends to other prizes.<ref>From [http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf "trails to Erdos"] {{webarchive|url=https://web.archive.org/web/20150924054224/http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf |date=2015-09-24 }}, 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 <nowiki>[</nowiki>Fields<nowiki>]</nowiki> 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).</ref> He did win the Wolf Prize, "for his numerous contributions to [[Number theory|number theory]], [[Combinatorics|combinatorics]], [[Probability|probability]], [[Set theory|set theory]] and [[Mathematical analysis|mathematical analysis]], and for personally stimulating mathematicians the world over".<ref>{{cite web |url=http://www.wolffund.org.il/cat.asp?id=23&cat_title=MATHEMATICS |title=Wolf Foundation Mathematics Prize Page |publisher=Wolffund.org.il |access-date=2010-05-29 |archive-url=https://web.archive.org/web/20080410234620/http://www.wolffund.org.il/cat.asp?id=23&cat_title=MATHEMATICS |archive-date=2008-04-10 |url-status=dead }}</ref> 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|probabilistic method]] especially stand out. [[Extremal combinatorics]] owes to him a whole approach, derived in part from the tradition of [[Analytic number theory|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 [[Selberg's identity|the first elementary proof]] for the [[Prime number theorem|prime number theorem]], along with [[Biography:Atle Selberg|Atle Selberg]]. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg.<ref name=goldfeld>{{Cite journal | last = Goldfeld | first = Dorian | year = 2003 | title = The Elementary Proof of the Prime Number Theorem: an Historical Perspective | journal = Number Theory: New York Seminar | pages = 179–192}}</ref><ref name=interview>{{Cite journal|url=https://www.ams.org/bull/2008-45-04/S0273-0979-08-01223-8/S0273-0979-08-01223-8.pdf |first1=Nils A.|last1= Baas|first2= Christian F.|last2= Skau |journal= Bull. Amer. Math. Soc. |volume=45 |year=2008|pages= 617–649 |title=The lord of the numbers, Atle Selberg. On his life and mathematics|doi=10.1090/S0273-0979-08-01223-8|issue=4|doi-access=free}}</ref> Erdős also contributed to fields in which he had little real interest, such as [[Topology|topology]], where he is credited as the first person to give an example of a totally disconnected [[Topological space|topological space]] that is not [[Zero-dimensional space|zero-dimensional]], the [[Erdős space]].<ref>{{cite web |url=http://www.maa.org/features/erdos.html |title=Reminiscences of Paul Erdös (1913–1996) |first=Melvin |last=Henriksen |publisher=Mathematical Association of America |access-date=2008-09-01}}</ref>

===Erdős's problems===
[[File:Paul Erdos with Terence Tao.jpg|thumb|Erdős influenced many young mathematicians. In this 1985 photo taken at the University of Adelaide, Erdős explains a problem to Terence Tao—who was 10 years old at the time. Tao received the Fields Medal in 2006, and was elected a Fellow of the Royal Society in 2007.]]
Erdős had a reputation for posing new problems as well as solving existing ones – [[Biography:Ernst G. Straus|Ernst Strauss]] called him "the absolute monarch of problem posers".<ref name="NYT_obit">{{cite news | title=Paul Erdos, 83, a Wayfarer In Math's Vanguard, Is Dead | author=Kolata, Gina | work={{wipe|The New York Times}} | pages=A1 and B8 | date=1996-09-24 |access-date=2008-09-29 |url=https://query.nytimes.com/gst/fullpage.html?res=9C02EED7113DF937A1575AC0A960958260 | author-link=Kolata, Gina}}</ref>  Throughout his career, Erdős would offer payments for solutions to unresolved problems.<ref>{{cite web |title=Math genius left unclaimed sum |url=http://www.edmontonjournal.com:80/business/Math+genius+left+unclaimed/3589345/story.html |website=Edmonton Journal |archive-url=https://web.archive.org/web/20110118231551/http://www.edmontonjournal.com/business/Math+genius+left+unclaimed/3589345/story.html |access-date=16 July 2020 |archive-date=2011-01-18 |url-status=live }}</ref> 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<ref>{{cite web |url=https://www.quantamagazine.org/mathematicians-prove-conjecture-on-big-prime-number-gaps-20141210/ |title=Prime Gap Grows After Decades-Long Lull|date=10 December 2014}}/</ref> 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 gap|prime gap]]s was solved in 2014, and the $10,000 paid.<ref>{{cite web |url=https://nautil.us/cash-for-math-the-erdos-prizes-live-on-10514/ |title=Cash for Math: The Erdős Prizes Live On |author=KEVIN HARTNETT |date=5 June 2017}}</ref>

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; [[Biography:Ronald Graham|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.<ref>{{cite journal |title=Erdös's Hard-to-Win Prizes Still Draw Bounty Hunters |first=Charles |last=Seife |journal=Science |doi=10.1126/science.296.5565.39 |date=5 April 2002 |volume=296 |pmid=11935003 |issue=5565 |pages=39–40|s2cid=34952867 }}</ref>{{Update inline|date=January 2024}}

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

{{blockquote|If the sum of the [[Multiplicative inverse|reciprocals]] of a [[Integer sequence|sequence of integers]] [[Divergent series|diverges]], then the sequence contains [[Arithmetic progression|arithmetic progression]]s 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 number|prime number]]s 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.<ref>p. 354, Soifer, Alexander (2008); ''[[The Mathematical Coloring Book|The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators]]''; New York: Springer. {{isbn|978-0-387-74640-1}}</ref>

The most familiar problem with an Erdős prize is likely the [[Collatz conjecture]], also called the 3''N''&nbsp;+&thinsp;1 problem. Erdős offered $500 for a solution.

===Collaborators===
Erdős' most frequent collaborators include Hungarian mathematicians [[Biography:András Sárközy|András Sárközy]] (62 papers) and [[Biography:András Hajnal|András Hajnal]] (56 papers), and American mathematician Ralph Faudree (50 papers). Other frequent collaborators were the following:<ref>[http://www.oakland.edu/enp/Erdos0p List of collaborators of Erdős by number of joint papers] {{webarchive|url=https://web.archive.org/web/20080804180523/http://www.oakland.edu/enp/Erdos0p |date=2008-08-04 }}, from the Erdős number project website.</ref>
{{colbegin|colwidth=18em}}
* Richard Schelp (42 papers)
* C. C. Rousseau (35 papers)
* Vera Sós (35 papers)
* [[Biography:Alfréd Rényi|Alfréd Rényi]] (32 papers)
* [[Biography:Pál Turán|Pál Turán]] (30 papers)
* [[Biography:Endre Szemerédi|Endre Szemerédi]] (29 papers)
* [[Biography:Ronald Graham|Ron Graham]] (28 papers)
* [[Biography:Stefan Burr|Stefan Burr]] (27 papers)
* [[Biography:Carl Pomerance|Carl Pomerance]] (23 papers)
* Joel Spencer (23 papers)
* [[Biography:János Pach|János Pach]] (21 papers)
* [[Biography:Miklós Simonovits|Miklós Simonovits]] (21 papers)
* [[Biography:Ernst G. Straus|Ernst G. Straus]] (20 papers)
* [[Biography:Melvyn B. Nathanson|Melvyn B. Nathanson]] (19 papers)
* [[Biography:Jean-Louis Nicolas|Jean-Louis Nicolas]] (19 papers)
* [[Biography:Richard Rado|Richard Rado]] (18 papers)
* [[Biography:Béla Bollobás|Béla Bollobás]] (18 papers)
* [[Biography:Eric Charles Milner|Eric Charles Milner]] (15 papers)
* [[Biography:András Gyárfás|András Gyárfás]] (15 papers)
* [[Biography:John Selfridge|John Selfridge]] (14 papers)
* [[Biography:Fan Chung|Fan Chung]] (14 papers)
* Richard R. Hall (14 papers)
* [[Biography:George Piranian|George Piranian]] (14 papers)
* István Joó (12 papers)
* Zsolt Tuza (12 papers)
* A. R. Reddy (11 papers)
* [[Biography:Vojtěch Rödl|Vojtěch Rödl]] (11 papers)
* Pál Révész (10 papers)
* [[Biography:Zoltán Füredi|Zoltán Füredi]] (10 papers)

{{colend}}
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|Erdős number}}
Because of his prolific output, friends created the Erdős number as a tribute. An Erdős number describes a person's degree of separation from Erdős himself, based on their collaboration with him, or with another who has their own Erdős number. 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,<ref>{{cite episode|url= http://www.wnyc.org/shows/radiolab/episodes/2009/10/09/segments/137643|title = From Benford to Erdös |series= Radio Lab|series-link= Radio Lab|airdate= 2009-09-30 |number= 2009-10-09|access-date = 2016-02-06|archive-url  = https://web.archive.org/web/20100818231149/http://www.wnyc.org/shows/radiolab/episodes/2009/10/09/segments/137643|archive-date = 2010-08-18|url-status     = dead}}</ref> 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 [[Social:Small-world experiment|small-world phenomenon]]). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics also have Erdős numbers.<ref>{{cite web |url=http://www.oakland.edu/enp/erdpaths/ |title=Some Famous People with Finite Erdös Numbers |first=Jerry |last=Grossman |access-date=2011-02-01}}</ref>

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.<ref name="trails">{{cite journal|last1=De Castro|first1=Rodrigo|last2= Grossman|first2= Jerrold W. |doi= 10.1007/BF03025416|issue=3|journal=[[The Mathematical Intelligencer]] |mr= 1709679|pages= 51–63 |title= Famous trails to Paul Erdős |url= http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf |volume=21|year=1999 |url-status= dead |archive-url =https://web.archive.org/web/20150924054224/http://www.oakland.edu/upload/docs/Erdos%20Number%20Project/trails.pdf |archive-date = 2015-09-24|citeseerx= 10.1.1.33.6972|s2cid = 120046886}} Original Spanish version in ''Rev. Acad. Colombiana Cienc. Exact. Fís. Natur.'' '''23''' (89) 563–582, 1999, {{MR|1744115}}.</ref> For example, the roughly 268,000 mathematicians with a known Erdős number have a median value of 5.<ref>{{Cite web|url=https://oakland.edu/enp/trivia/index|title=Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University|website=OU-Main-Page|access-date=October 6, 2019}}</ref> In contrast, the median Erdős number of Fields Medalists is 3.<ref>{{cite web |url=http://www.financial-math.org/blog/2016/11/erdos-numbers-in-finance/ |title=Erdös Numbers in Finance}}</ref> As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or less.<ref>{{cite web |url=https://files.oakland.edu/users/grossman/enp/Erdos2.html |title=Erdos2}}</ref><ref name="oakland.edu">{{Cite web|url=https://oakland.edu/enp/erdpaths/index|title=Paths to Erdös - The Erdös Number Project- Oakland University|website=OU-Main-Page|access-date=October 6, 2019}}</ref> Collaboration distances will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the [[Mathematical Reviews]] catalogue.<ref name="ams">{{cite web|url=https://www.ams.org/mathscinet/collaborationDistance.html|website=ams.org|title=mathscinet/collaborationDistance|access-date=2017-12-02}}</ref>

The Erdős number was most likely first defined by Casper Goffman,<ref>{{Cite web|author=Michael Golomb |title=Obituary of Paul Erdös at Purdue |url=https://www.math.purdue.edu/about/purview/fall96/paul-erdos.html|access-date=2022-05-04|website=www.math.purdue.edu}}</ref> an [[Mathematical analysis|analyst]] whose own Erdős number is 2.<ref>[https://files.oakland.edu/users/grossman/enp/ErdosA.html from the Erdos Number Project]</ref> Goffman published his observations about Erdős's prolific collaboration in a 1969 article titled "And what is your Erdős number?"<ref>{{cite journal|author=Goffman, Casper|title=And what is your Erdős number?|jstor=2317868|journal=American Mathematical Monthly|volume=76|issue=7|pages=791|year=1969|doi=10.2307/2317868}}</ref>

Jerry Grossman has written that it could be argued that Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for [[Biography:Carl Pomerance|Carl Pomerance]]) when [[Organization:Emory University|Emory University]] awarded them honorary degrees on the same day.<ref>{{cite web |url=http://www.oakland.edu/enp/related/ |title=Items of Interest Related to Erdös Numbers |first=Jerry|last=Grossman}}</ref> Erdős numbers have also been proposed for an infant, a horse, and several actors.<ref name="harveycohen">{{cite web|url=http://harveycohen.net/erdos/|website=harveycohen.net|title=The Extended Erdös Number Project |access-date=2017-12-02}}</ref>

==Personality==
{{quote box
| align = left
| quote = Another roof, another proof.
| source = — Paul Erdős<ref>{{Cite book|isbn=9789814723930|page=293|editor-last1=Chern|editor-first1=Shiing-Shen|editor-last2=Hirzebruch|editor-first2=Friedrich|title=Wolf Prize in Mathematics|date=2 September 2023 |volume=1|publisher=World Scientific }}</ref>
}}

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 traveling between scientific conferences, universities and the homes of colleagues all over the world. He earned enough in stipends from universities as a guest lecturer, and from various mathematical awards, to fund his travels and basic needs; money left over he used to fund cash prizes for proofs of "Erdős's problems" (see above). 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 to visit next.

His colleague [[Biography:Alfréd Rényi|Alfréd Rényi]] said, "a mathematician is a machine for turning coffee into theorems",<ref>{{Cite web|title=Biography of Alfréd Rényi |author1=J.J. O'Connor |author2=E.F. Robertson |url=https://mathshistory.st-andrews.ac.uk/Biographies/Renyi/|access-date=2022-05-04|website=Maths History |date=December 2008 |language=en}}</ref> and Erdős drank copious quantities; this quotation is often attributed incorrectly to Erdős,{{sfn|Schechter|1998|pp=155}} but Erdős himself ascribed it to Rényi.<ref>{{cite journal |first=Paul |last=Erdős |title=Child Prodigies |journal=Mathematics Competitions |volume=8 |url=http://www.amt.edu.au/mc19951erdos.pdf |access-date=2012-07-17 |number=1 |year=1995 |pages=7–15 |url-status=dead |archive-url=https://web.archive.org/web/20120324221556/http://www.amt.edu.au/mc19951erdos.pdf |archive-date=2012-03-24 }}</ref> After his mother's death in 1971 he started taking antidepressants and amphetamines, despite the concern of his friends, one of whom ([[Biography:Ronald Graham|Ron Graham]]) bet him $500 that he could not stop taking them for a month. Erdős won the bet, but complained that it impacted his performance: "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathematics back a month."<ref>Hill, J. [http://www.untruth.org/~josh/math/Paul%20Erd%F6s%20bio-rev2.pdf Paul Erdos, Mathematical Genius, Human (In That Order)]</ref> After he won the bet, he promptly resumed his use of Ritalin and Benzedrine.<ref name=":3"/>

He had his own idiosyncratic vocabulary; although an agnostic atheist,<ref>{{cite web|title=Centenary of Mathematician Paul Erdős – Source of Bacon Number Concept|url=http://www.huffingtonpost.com/colm-mulcahy/paul-erdos_b_2945871.html|work=Huffington Post|access-date=2013-04-13|first=Colm|last=Mulcahy|date=2013-03-26|quote=In his own words, "I'm not qualified to say whether or not God exists. I kind of doubt He does. Nevertheless, I'm always saying that the SF has this transfinite Book that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect...You don't have to believe in God, but you should believe in the Book.".|author-link=Colm Mulcahy}}</ref><ref>{{cite book|title=Quotable Atheist: Ammunition for Nonbelievers, Political Junkies, Gadflies, and Those Generally Hell-Bound|year=2008|publisher=Nation Books|isbn=9781568584195|page=107|first=Jack|last=Huberman|quote=I kind of doubt He [exists]. Nevertheless, I'm always saying that the SF has this transfinite Book ... that contains the best proofs of all theorems, proofs that are elegant and perfect.... You don't have to believe in God, but you should believe in the Book.}}</ref> he spoke of "The Book", a visualization of a book in which [[Religion:God|God]] had written down the best and most elegant proofs for mathematical theorems.<ref>{{cite book|title=Mathematics and the Aesthetic: New Approaches to an Ancient Affinity|year=2006|publisher=Springer|isbn=9780387305264|editor=[[Biography:Nathalie Sinclair|Nathalie Sinclair]], William Higginson|page=36|quote=Erdös, an atheist, named 'the Book' the place where God keeps aesthetically perfect proofs.}}</ref> 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).{{sfn|Schechter|1998|pp=70–71}}<ref>{{cite book|title=Variety in Religion And Science: Daily Reflections|year=2005|publisher=iUniverse|isbn=9780595358403|first=Varadaraja|last=Raman|page=256}}</ref> He accused SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly [[Mathematical beauty|beautiful mathematical proof]] he would exclaim, "This one's from ''The Book''!" This later inspired a book titled ''[[Proofs from THE BOOK|Proofs from the Book]]''.

Other idiosyncratic elements of Erdős's vocabulary include:<ref name=":3">{{Cite book|last=Paul |first=Hoffman |title=The Man Who Loved Only Numbers |url=https://archive.nytimes.com/www.nytimes.com/books/first/h/hoffman-man.html |chapter=1. The Story of Paul Erdös and the Search for Mathematical Truth |access-date=2022-05-04}}</ref>
* Children were referred to as "epsilons" (because in mathematics, particularly [[Calculus|calculus]], an arbitrarily small positive quantity is commonly denoted by the Greek letter (ε)).
* Women were "bosses" who "captured" men as "slaves" by marrying them. Divorced men were "liberated".
* People who stopped doing mathematics had "died", while people who died had "left".
* Alcoholic drinks were "poison".
* Music (except classical music) was "noise".
* To be considered a hack was to be a "Newton".
* To give a mathematical lecture was "to preach".
*Mathematical lectures themselves were "sermons".<ref>{{Cite web|last=Strick|first=Heinz|title=Paul Erdős|url=https://mathshistory.st-andrews.ac.uk/Strick/Erdos.pdf}}</ref>
* To give an oral exam to students was "to torture" them.

He gave nicknames to many countries, examples being: the U.S. was "samland" (after Uncle Sam)<ref name=":3"/> and the Soviet Union was "joedom" (after {{wipe|Joseph Stalin}}).<ref name=":3"/>  He claimed that [[Social:Hindi|Hindi]] was the best language because words for old age (''bud̩d̩hā'') and stupidity (''buddhū'') sounded almost the same.{{sfn|Bollobás|1996|pp=6}}

=== Signature ===
Erdős signed his name "Paul Erdos P.G.O.M." When he became 60, he added "L.D.", at 65 "A.D.", at 70 "L.D." (again), and at 75 "C.D."{{sfn|Bollobás|1996|pp=6}}
* P.G.O.M. represented "Poor Great Old Man"
* The first L.D. represented "Living Dead"
* A.D. represented "Archaeological Discovery"
* The second L.D. represented "Legally Dead"
* C.D. represented "Counts Dead"{{sfn|Schechter|1998|pp=41}}<ref>{{YouTube|zRNGV85kPbI|Paul Erdös: N is a number}}, a documentary film by George Paul Csicsery, 1991.</ref>

==Legacy==
[[File:Erdős Pál sírja.jpg|thumb|Grave of Erdős, Kozma Street Cemetery, Budapest]]
=== Books and films ===
Erdős is the subject of at least three books: two biographies (Hoffman's ''The Man Who Loved Only Numbers'' and Schechter's ''My Brain is Open'', both published in 1998) and a 2013 children's picture book by Deborah Heiligman (''The Boy Who Loved Math: The Improbable Life of Paul Erdős'').<ref>{{cite news|last=Silver|first=Nate|title=Children's Books Beautiful Minds 'The Boy Who Loved Math' and 'On a Beam of Light'|url=https://www.nytimes.com/2013/07/14/books/review/the-boy-who-loved-math-and-on-a-beam-of-light.html?_r=0|access-date=2014-10-29|work=The New York Times|date=12 July 2013}}</ref>

He is also the subject of [[Biography:George Csicsery|George Csicsery]]'s biographical documentary film ''N is a Number: A Portrait of Paul Erdős,''<ref>{{Citation|last=Csicsery|first=George Paul|title=N Is a Number: A Portrait of Paul Erdös|url=https://www.imdb.com/title/tt0125425/?ref_=nv_sr_srsg_0|type=Documentary, Biography|access-date=2022-05-04}}</ref> made while he was still alive.

===Astronomy===
In 2021 the [[Astronomy:Minor planet|minor planet]] ([[Astronomy:Asteroid|asteroid]]) 405571 (temporarily designated 2005 QE87) was formally named "Erdőspál" to commemorate Erdős, with the citation describing him as "a Hungarian mathematician, much of whose work centered around discrete mathematics. His work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics."<ref>{{cite journal |last1=Working Group on Small Body Nomenclature of the International Astronomical Union |date=2021-05-14 |title=WGSBN Bulletin |url=https://www.wgsbn-iau.org/files/Bulletins/V001/WGSBNBull_V001_001.pdf |journal=WGSBN Bulletin |volume=1 |issue=1 |pages=29  |access-date=2021-05-16}}</ref> The naming was proposed by "K. Sárneczky, Z. Kuli" (Kuli being the asteroid's discoverer).

==See also==
* {{Annotated link|List of topics named after Paul Erdős}}&nbsp;– including conjectures, numbers, prizes, and theorems
* {{Annotated link|Box-making game}}
* {{Annotated link|Covering system}}
* {{Annotated link|Dimension (graph theory)}}
* {{Annotated link|Even circuit theorem}}
* {{Annotated link|Friendship graph}}
* {{Annotated link|Minimum overlap problem}}
* {{Annotated link|Probabilistic method}}
* {{Annotated link|Probabilistic number theory}}
* {{Annotated link|The Martians (scientists)}}

==References==
{{Reflist|30em}}

==Sources==
* {{Cite book|title=Math and mathematicians : the history of math discoveries around the world|last=Bruno|first=Leonard C. |year=2003|orig-year=1999|publisher=U X L|others=Baker, Lawrence W.|isbn=978-0787638139|location=Detroit, Mich.|oclc=41497065|url=https://archive.org/details/mathmathematicia00brun|url-access=registration}}
* {{cite book | first=Bruce | last=Schechter | title=My Brain is Open: The Mathematical Journeys of Paul Erdős | publisher=Simon & Schuster |location=New York |year=1998 | isbn=978-0-684-84635-4 | url=https://archive.org/details/mybrainisopenmat00sche }}
*{{cite journal |last1=Bollobás |first1=Béla |title=A Life of Mathematics – Paul Erdos, 1913-1996 |journal=Focus |date=December 1996 |volume=16 |issue=6 |page=4 |url=https://www.maa.org/sites/default/files/pdf/pubs/focus/past_issues/FOCUS_16_6.pdf |access-date=6 May 2022 |publisher=Mathematical Association of America |location=Washington, D.C. |language=en}}
* {{cite book | last=Hoffman | first=Paul | title=The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth | publisher=Fourth Estate Ltd | location=London | year=1998 | isbn=978-1-85702-811-9 |  title-link=The Man Who Loved Only Numbers }}

==Further reading==
* {{cite book |title=Proofs from THE BOOK |last1=Aigner |first1=Martin |first2=Günther |last2=Ziegler |year= 2014|publisher=Springer |location=Berlin; New York |isbn=978-3-662-44204-3 |doi=10.1007/978-3-662-44205-0|title-link=Proofs from THE BOOK }}

==External links==
* [https://scholar.google.com.au/citations?hl=en&user=_sb8TE0AAAAJ&view_op=list_works&pagesize=100 Erdős's Google Scholar profile]
* [http://www.renyi.hu/~p_erdos/ Searchable collection of (almost) all papers of Erdős]
* {{MacTutor Biography|id=Erdos}}
* {{MathGenealogy|19470}}
* Jerry Grossman at Oakland University. [http://www.oakland.edu/enp/ ''The Erdös Number Project'']
* [https://web.archive.org/web/20050419081411/http://www.vega.org.uk/series/vpl/vpl1/index.php The Man Who Loved Only Numbers] – Royal Society public lecture by Paul Hoffman (video)
* [https://web.archive.org/web/20090928053424/http://www.wnyc.org/shows/radiolab/episodes/2009/10/09 Radiolab: Numbers, with a story on Paul Erdős]
* [http://math.ucsd.edu/~fan/ep.pdf Fan Chung, "Open problems of Paul Erdős in graph theory"]

{{Wolf Prize in Mathematics}}

{{DEFAULTSORT:Erdos, Paul}}
[[Category:Combinatorialists]]
[[Category:Set theorists]]
[[Category:Number theorists]]
[[Category:Probability theorists]]

{{Sourceattribution|Paul Erdős}}