Paul Erdős (Hungarian:Erdős Pál[ˈɛrdøːʃˈpaːl]; 26March 1913 – 20September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematicalconjectures[2] of the 20th century.[3]Erdős pursued and proposed problems indiscrete mathematics,graph theory,number theory,mathematical analysis,approximation theory,set theory, andprobability theory.[4] Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed toRamsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previouslyopen problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed.[5]
He was known both for his social practice of mathematics, working with more than 500 collaborators, and for hiseccentric lifestyle;Time magazine called him "The Oddball's Oddball".[6] 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 devoted his waking hours to mathematics, even into his later years; he died at a mathematics conference inWarsaw in 1996.[7]
Erdős's prolific output with co-authors prompted the creation of theErdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.
Paul Erdős was born on 26 March 1913, inBudapest,Austria-Hungary,[8] the only surviving child of Anna (née Wilhelm) and Lajos Erdős (né Engländer).[9][10] His two sisters, aged three and five, both died ofscarlet fever a few days before he was born.[11] His parents, bothJewish, were high school mathematics teachers. His fascination with mathematics developed early. He was raised partly by a German governess[12] because his father was held captive inSiberia as an Austro-Hungarianprisoner of war during 1914–1920,[10] 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.[13]
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.[12] 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.[14][15]
When he was 16, his father introduced him to two subjects that would become lifetime favourites—infinite series andset theory. In high school, Erdős became an ardent solver of the problems that appeared each month inKöMaL, the "Mathematical and Physical Journal for Secondary Schools".[16]
Because he was Jewish, Hungary was dangerous so he left the country, relocating to the United States in 1938.[17] 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 theInstitute for Advanced Study in Princeton at the time.[17][18] 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".[13]
Described by his biographer, Paul Hoffman, as "probably the most eccentric mathematician in the world," Erdős spent most of his adult lifeliving out of a suitcase.[19] 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.[19] 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.[19]
Ulam left his post at theUniversity of Wisconsin–Madison in 1943 to work on theManhattan Project inLos 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.[13]
On 20 September 1996, at the age of 83, he had aheart attack and died while attending a conference inWarsaw.[20] These circumstances were close to the way he wanted to die. He once said,
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.[20]
Erdős never married and had no children.[9] He is buried next to his mother and father in the JewishKozma Street Cemetery in Budapest.[21] For hisepitaph, he suggested "I've finally stopped getting dumber." (Hungarian:"Végre nem butulok tovább").[22]
Erdős's name contains the Hungarian letter "ő" ("o" withdouble acute accent), but is often incorrectly written asErdos orErdös either "by mistake or out of typographical necessity".[23]
In 1934, Erdős moved toManchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at theInstitute for Advanced Study, Princeton, New Jersey, for the next ten years. Despite outstanding papers withMark Kac andAurel Wintner onprobabilistic number theory,Pál Turán in approximation theory, andWitold Hurewicz on dimension theory, his fellowship was not continued, andErdős was forced to take positions as a wandering scholar atUPenn,Notre Dame,Purdue,Stanford, andSyracuse.[24] He would not stay long in one place, instead traveling among mathematical institutions until his death.
As a result of theRed Scare andMcCarthyism,[25][26][27] in 1954, theUnited States Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States.[28] Teaching at theUniversity 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. He was given a position for three months at the Hebrew University in Jerusalem, and then a "permanent visiting professor" position at the Technion.
Hungary at the time was under theWarsaw Pact with theSoviet 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 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.[29]
Erdős was one of the most prolific publishers of papers in mathematical history, comparable only withLeonhard Euler; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by himself.[34] Erdős wrote around 1,525 mathematical articles in his lifetime,[35] mostly with co-authors. He strongly believed in and practiced mathematics as a social activity,[36] having 511 different collaborators in his lifetime.[37]
In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics"[38] byTimothy 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 particulartheorems and conjectures throughout his illustrious career."[39] Erdős never won theFields Medal (the highest mathematical prize available during his lifetime), nor did he coauthor a paper with anyone who did,[40] a pattern that extends to other prizes.[41] He did win the 1983/84Wolf Prize, "for his numerous contributions tonumber theory,combinatorics,probability,set theory andmathematical analysis, and for personally stimulating mathematicians the world over".[42] In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".
Erdős had a reputation for posing new problems as well as solving existing ones:Ernst Strauss called him "the absolute monarch of problem posers".[7] Throughout his career, Erdős would offer payments for solutions to unresolved problems.[46] 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[47] 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 onprime gaps was solved in 2014, and the $10,000 paid.[48]
There are thought to be at least a thousand remaining unsolved problems, though there is no official or comprehensive list. The offers remained active despite Erdős's death;Ronald Graham was the (informal) administrator of solutions, and a solver could receive either an original check signed by Erdős before his death (for memento only, which cannot be cashed) or a cashable check from Graham.[49][needs update] British mathematicianThomas Bloom started a website dedicated to Erdős's problems in 2024.[50]
If true, it would solve several other open problems in number theory, although one main implication of the conjecture, that theprime numbers contain arbitrarily long arithmetic progressions, has since been proved independently as theGreen–Tao theorem. The payment for the solution of the problem is currently worth US$5,000.[51]
The most familiar problem with an Erdős prize is likely theCollatz conjecture, also called the 3N + 1 problem. Erdős offered $500 for a solution.[52]
Erdős's most frequent collaborators include Hungarian mathematiciansAndrás Sárközy (62 papers) andAndrás Hajnal (56 papers), and American mathematicianRalph Faudree (50 papers). Other frequent collaborators were the following:[53]
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,[54] 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 thesmall-world phenomenon). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics also have Erdős numbers.[55]
Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.[56] For example, the roughly 268,000 mathematicians with a known Erdős number have a median value of 5.[57] In contrast, the median Erdős number ofFields Medalists is 3.[58] As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or lower.[59][60] Collaboration distances will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration. TheAmerican Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in theMathematical Reviews catalogue.[61]
The Erdős number was most likely first defined by Casper Goffman,[62] ananalyst whose own Erdős number is 2; Goffman co-authored with mathematicianRichard B. Darst, who co-authored with Erdős.[63] Goffman published his observations about Erdős's prolific collaboration in a 1969 article titled "And what is your Erdős number?"[64]
Jerry Grossman has written that it could be argued thatBaseball Hall of FamerHank Aaron can be considered to have an Erdős number of 1, because they both autographed the same baseball forCarl Pomerance whenEmory University awarded them honorary degrees on the same day.[65] Erdős numbers have also been proposed for an infant, a horse, and several actors.[66]
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 generallydonated 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 colleagueAlfréd Rényi said, "A mathematician is a machine for turningcoffee into theorems",[68] and Erdős drank copious quantities; this quotation is often attributed incorrectly to Erdős,[69] but Erdős ascribed it to Rényi.[70] After his mother's death in 1971 he started taking antidepressants and amphetamines, despite the concern of his friends, one of whom (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."[71] After he won the bet, he promptly resumed his use ofRitalin andBenzedrine.[72]
He had his own idiosyncratic vocabulary; although anagnostic atheist,[73][74] he spoke of "The Book", a visualization of a book in whichGod had written down the best and most elegant proofs for mathematical theorems.[75] Lecturing in 1985 he said, "You don't have to believe in God, but you should believe inThe Book." He doubted the existence of God.[76][77] He playfully nicknamed him the SF (for "Supreme Fascist"), accusing him of hiding his socks and Hungarianpassports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularlybeautiful mathematical proof he would exclaim, "This one's fromThe Book!" This later inspired a book titledProofs from the Book.
Other idiosyncratic elements of Erdős's vocabulary include:[72]
Children were referred to as "epsilons", because in mathematics, particularlycalculus, 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".
To give an oral exam to students was "to torture" them.
He gave nicknames to many countries, examples being: the U.S. was "samland" (afterUncle Sam)[72] and the Soviet Union was "joedom" (afterJoseph Stalin).[72] He claimed thatHindi was the best language, because words for old age (bud̩d̩hā) and stupidity (buddhū) sounded almost the same.[79]
Erdős is the subject of at least three books: two biographies (Hoffman'sThe Man Who Loved Only Numbers and Schechter'sMy Brain is Open, both published in 1998) and a 2013 children's picture book byDeborah Heiligman (The Boy Who Loved Math: The Improbable Life of Paul Erdős).[82]
In 2021 theminor planet (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."[84] The naming was proposed by "K. Sárneczky, Z. Kuli" (Kuli being the asteroid's discoverer).
In 2025 British arachnologists Danniella Sherwood and R. C. Gallon describedHeterothele erdosi Sherwood & Gallon, 2025, a new species oftarantula fromNigeria, named in honour of Erdős.[85]
^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.", fromErdős, Paul; Miklós, D.;Sós, Vera T. (1996).Combinatorics, Paul Erdős is eighty.
^Baker, A.; Bollobás, B. (1999). "Paul Erdõs. 26 March 1913 — 20 September 1996".Biographical Memoirs of Fellows of the Royal Society.45. The Royal Society:147–164.doi:10.1098/rsbm.1999.0011.ISSN0080-4606.S2CID123517792.
^Erdős, Paul (4 June 1996)."Dear President Downey"(PDF). Archived fromthe original(PDF) on 15 October 2005. Retrieved8 July 2014.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.'
^Gowers, Timothy (2000)."The Two Cultures of Mathematics"(PDF). In Arnold, V. I.; Atiyah, Michael; Lax, Peter D.; Mazur, Barry (eds.).Mathematics: Frontiers and Perspectives. American Mathematical Society.ISBN978-0821826973.
^Spencer, Joel (November–December 2000). "Prove and Conjecture!".American Scientist.88 (6). This article is a review ofMathematics: Frontiers and Perspectives
^From"trails to Erdos"Archived 2015-09-24 at theWayback Machine, by DeCastro and Grossman, inThe 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).
^Erdős, Paul (1995)."Child Prodigies"(PDF).Mathematics Competitions.8 (1):7–15. Archived fromthe original(PDF) on 24 March 2012. Retrieved17 July 2012.
^abcdPaul, Hoffman. "1. The Story of Paul Erdös and the Search for Mathematical Truth".The Man Who Loved Only Numbers. Retrieved4 May 2022.
^Mulcahy, Colm (26 March 2013)."Centenary of Mathematician Paul Erdős – Source of Bacon Number Concept".Huffington Post. Retrieved13 April 2013.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.".
^Huberman, Jack (2008).Quotable Atheist: Ammunition for Nonbelievers, Political Junkies, Gadflies, and Those Generally Hell-Bound. Nation Books. p. 107.ISBN9781568584195.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.
^Nathalie Sinclair, William Higginson, ed. (2006).Mathematics and the Aesthetic: New Approaches to an Ancient Affinity. Springer. p. 36.ISBN9780387305264.Erdös, an atheist, named 'the Book' the place where God keeps aesthetically perfect proofs.
^Sherwood, D., Gallon, R. C. (2025). "Heterotheleerdosi, a new species ofHeterothele Karsch, 1879 from Nigeria (Araneae: Theraphosidae)".Natural Somogyiensis 45: 51–56.[1]
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