53-680: Erdős , Erdos , or Erdoes is a Hungarian surname. Paul Erdős (1913–1996), was a Hungarian mathematician. Paul Erd%C5%91s Paul Erdős ( Hungarian : Erdős Pál [ˈɛrdøːʃ ˈpaːl] ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. Erdős pursued and proposed problems in discrete mathematics , graph theory , number theory , mathematical analysis , approximation theory , set theory , and probability theory . Much of his work centered around discrete mathematics, cracking many previously unsolved problems in
106-548: A 1969 article titled "And what is your Erdős number?" 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 Carl Pomerance ) when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been proposed for an infant, a horse, and several actors. Another roof, another proof. — Paul Erdős Possessions meant little to Erdős; most of his belongings would fit in
159-425: A certain section of Budapest middle-class society, many members of which had the same manners, if not quite the same mannerisms, as Fejér — there he would appear about half eccentric. Pólya writes the following about Fejér, telling us much about his personality: He had artistic tastes. He deeply loved music and was a good pianist. He liked a well-turned phrase. 'As to earning a living', he said, 'a professor's salary
212-553: 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 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, but Erdős himself ascribed it to Rényi. After his mother's death in 1971 he started taking antidepressants and amphetamines, despite
265-638: A few days before he was born. His parents, both Jewish , were high school mathematics teachers. His fascination with mathematics developed early. He was raised partly by a German governess because his father was held captive in Siberia as an Austro-Hungarian prisoner of war during 1914–1920, causing his mother to have to work long hours to support their household. His father had taught himself English while in captivity, but mispronounced many words. When Lajos later taught his son to speak English, Paul learned his father's pronunciation, which he continued to use for
318-519: A free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue. The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 2; Goffman co-authored with mathematician Richard B. Darst , who co-authored with Erdős. Goffman published his observations about Erdős's prolific collaboration in
371-604: A mathematics conference in Warsaw . 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. Paul Erdős was born on 26 March 1913, in Budapest , Austria-Hungary , the only surviving child of Anna ( née Wilhelm) and Lajos Erdős (né Engländer). His two sisters, aged three and five, both died of scarlet fever
424-448: A proof for Bertrand's postulate which proved to be far neater than Chebyshev 's original one. He also discovered the first elementary proof for the prime number theorem , along with Atle Selberg . However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg. Erdős also contributed to fields in which he had little real interest, such as topology , where he
477-410: A social activity, having 511 different collaborators in his lifetime. In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics" by Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated). Joel Spencer states that "his place in the 20th-century mathematical pantheon
530-475: A social activity, living an itinerant lifestyle (i.e., someone whose way of life involves travelling around) 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". He devoted his waking hours to mathematics, even into his later years—indeed, his death came at
583-458: A strong mathematical school: he has educated a new generation of students who have gone on to become eminent scientists. As Polya recalled, a large number of them became interested in mathematics thanks to Fejér, his fascinating personality and charisma. Fejér gave short (no more than an hour) but very entertaining lectures and often sat with students in cafés, discussing mathematical problems and telling stories from his life and how he interacted with
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#1732791781798636-539: 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
689-451: 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 ). He is also the subject of George Csicsery 's biographical documentary film N is a Number: A Portrait of Paul Erdős , made while he was still alive. In 2021 the minor planet ( asteroid ) 405571 (temporarily designated 2005 QE87) was formally named "Erdőspál" to commemorate Erdős, with
742-603: Is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career." Erdős never won the Fields Medal (the highest mathematical prize available during his lifetime), nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the 1983/84 Wolf Prize , "for his numerous contributions to number theory , combinatorics , probability , set theory and mathematical analysis , and for personally stimulating mathematicians
795-499: Is a necessary, but not sufficient, condition.' Once he was very angry with a colleague who happened to be a topologist , and explaining the case at length he wound up by declaring '... and what he is saying is a topological mapping of the truth'. He had a quick eye for foibles and miseries; in seemingly dull situations he noticed points that were unexpectedly funny or unexpectedly pathetic. He carefully cultivated his talent of raconteur; when he told, with his characteristic gestures, of
848-459: Is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional , the Erdős space . Erdős had a reputation for posing new problems as well as solving existing ones – Ernst Strauss called him "the absolute monarch of problem posers". Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from $ 25 for problems that he felt were just out of
901-412: 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, cannot be cashed) or a cashable check from Graham. British mathematician Thomas Bloom started a website dedicated to Erdős's problems in 2024. Perhaps
954-590: The Green–Tao theorem ). The payment for the solution of the problem is currently worth US$ 5,000. The most familiar problem with an Erdős prize is likely the Collatz conjecture , also called the 3 N + 1 problem. Erdős offered $ 500 for a solution. Erdős' 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
1007-562: The University of Waterloo over what he considered to be unfair treatment of colleague Adrian Bondy . Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler ; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by himself. Erdős wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as
1060-588: The 60-year-old Erdős voluntarily left Hungary. 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 . He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1977. Shortly before his death, he renounced his honorary degree from
1113-675: 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, 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 ). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics also have Erdős numbers. Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. For example,
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#17327917817981166-708: 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". 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 Mark Kac and Aurel Wintner on probabilistic number theory , Pál Turán in approximation theory, and Witold Hurewicz on dimension theory, his fellowship
1219-620: The Technion. Hungary at the time was under the Warsaw Pact with the Soviet Union . Although Hungary limited the freedom of its own citizens to enter and exit the country, in 1956 it gave Erdős the exclusive privilege of being allowed to enter and exit the country as he pleased. 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,
1272-579: The U.S. was "samland" (after Uncle Sam ) and the Soviet Union was "joedom" (after Joseph Stalin ). He claimed that Hindi was the best language because words for old age ( bud̩d̩hā ) and stupidity ( buddhū ) sounded almost the same. 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." Erdős is the subject of at least three books: two biographies ( Hoffman 's The Man Who Loved Only Numbers and Schechter's My Brain
1325-667: The United States. 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
1378-460: The age of 21, he was awarded a doctorate in mathematics. Erdős's thesis advisor was Lipót Fejér , who was also the thesis advisor for John von Neumann , George Pólya , and Paul (Pál) Turán . He took up a post-doctoral fellowship at Manchester , as Jews in Hungary were suffering oppression under the authoritarian regime . While there he met Godfrey Harold Hardy and Stan Ulam . Because he
1431-577: 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. Erdős never married and had no children. He is buried next to his mother and father in the Jewish Kozma Street Cemetery in Budapest. For his epitaph , he suggested "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább" ). Erdős's name contains
1484-512: The bet, he promptly resumed his use of Ritalin and Benzedrine . He had his own idiosyncratic vocabulary; although an agnostic atheist , he spoke of "The Book", a visualization of a 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
1537-482: 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." The naming was proposed by "K. Sárneczky, Z. Kuli" (Kuli being the asteroid's discoverer). Lip%C3%B3t Fej%C3%A9r Lipót Fejér (or Leopold Fejér , Hungarian pronunciation: [ˈfɛjeːr] ; 9 February 1880 – 15 October 1959)
1590-416: 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." After he won
1643-534: The end of December 1944, members of the Arrow Cross Party stormed into his house. Fejér and all the residents of his house were convoyed to the banks of the Danube and were about to be shot , but were miraculously saved by a phone call "from a brave officer". Fejér was later found in a hospital in the city, where he was admitted "under unexplained circumstances". This severe trauma left a permanent mark on
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1696-589: The existence of God. He playfully nicknamed him the SF (for "Supreme Fascist"), accusing him 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 titled Proofs from the Book . Other idiosyncratic elements of Erdős's vocabulary include: He gave nicknames to many countries, examples being:
1749-411: The expected year due to Erdős not conforming to the standards of the place; they found him "uncouth and unconventional". 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 . Except for some years in the 1950s, when he was not allowed to enter the United States based on the accusation that he
1802-401: The field. He championed and contributed to Ramsey theory , which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems , rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be
1855-449: The following: For other co-authors of Erdős, see the list of people with Erdős number 1 in List of people by 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
1908-441: The intuitively clear detail. It was not given to him to solve very difficult problems or to build vast conceptual structures. Yet he could perceive the significance, the beauty, and the promise of a rather concrete not too large problem, foresee the possibility of a solution and work at it with intensity. And, when he had found the solution, he kept on working at it with loving care, till each detail became fully transparent. It
1961-405: The little shortcomings of a certain great mathematician, he was irresistible. The hours spent in continental coffee houses with Fejér discussing mathematics and telling stories are a cherished recollection for many of us. Fejér presented his mathematical remarks with the same verve as his stories, and this may have helped him in winning the lasting interest of so many younger men in his problems. In
2014-485: The most mathematically 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
2067-473: The problems that appeared each month in KöMaL , the "Mathematical and Physical Journal for Secondary Schools". Erdős began studying at the University of Budapest when he was 17 after winning a national examination. At the time, admission of Jews to Hungarian universities was severely restricted under the numerus clausus . By the time he was 20, he had found a proof for Chebyshev's theorem . In 1934, at
2120-404: The project, but the invitation was withdrawn when Erdős expressed a desire to return to Hungary after the war. On 20 September 1996, at the age of 83, he had a heart attack and died while attending a conference in Warsaw . 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
2173-408: The reach of the current mathematical thinking (both his and others) up to $ 10,000 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 gaps was solved in 2014, and the $ 10,000 paid. There are thought to be at least a thousand remaining unsolved problems, though there
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2226-565: The rest of his life. He taught himself to read through mathematics texts that his parents left around in their home. By the age of five, given a person's age, he could calculate in his head how many seconds they had lived. Due to his sisters' deaths, he had a close relationship with his mother, with the two of them reportedly sharing the same bed until he left for college. When he was 16, his father introduced him to two subjects that would become lifetime favourites— infinite series and set theory . In high school, Erdős became an ardent solver of
2279-443: The roughly 268,000 mathematicians with a known Erdős number have a median value of 5. In contrast, the median Erdős number of Fields Medalists is 3. As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or less. 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
2332-420: The same article Pólya writes about Fejér's style of mathematics: Fejér talked about a paper he was about to write up. 'When I write a paper,' he said, 'I have to rederive for myself the rules of differentiation and sometimes even the commutative law of multiplication.' These words stuck in my memory and years later I came to think that they expressed an essential aspect of Fejér's mathematical talent; his love for
2385-518: The scientist's mental faculties, something even he himself noticed and later often said of himself "since I became an idiot". Still, according to his colleagues, he kept on an even keel until mid-1950s, when he became senile. Lipót Fejér died in Budapest on 15 October 1959. His grave is in the distinguished Kerepesi Cemetery . If you could see him in his rather Bohemian attire (which was, I suspect, carefully chosen) you would find him very eccentric. Yet he would not appear so in his natural habitat, in
2438-442: The world over". 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
2491-514: The world's leading mathematicians. Fejér's research concentrated on harmonic analysis and, in particular, Fourier series . Fejér collaborated to produce important papers, one with Carathéodory on entire functions in 1907 and another major work with Frigyes Riesz in 1922 on conformal mappings (specifically, a short proof of the Riemann mapping theorem ). In 1944, Fejér was forced to resign because of his Jewish background. One night at
2544-581: Was Jewish, Erdős decided Hungary was dangerous and left the country, relocating to the United States in 1938. 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 Institute for Advanced Study in Princeton at the time. However, his fellowship at Princeton only got extended by 6 months rather than
2597-620: Was a Communist sympathizer, his life was a continuous series of going from one meeting or seminar to another. During his visits, Erdős expected his hosts to lodge him, feed him, and do his laundry, along with anything else he needed, as well as arrange for him to get to his next destination. Ulam left his post at the University of Wisconsin–Madison in 1943 to work on the Manhattan Project in Los Alamos, New Mexico with other mathematicians and physicists. He invited Erdős to join
2650-528: Was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz , and changed to the Hungarian name Fejér around 1900. He was born in Pécs , Austria-Hungary , into the Jewish family of Victoria Goldberger and Samuel Weiss. His maternal great-grandfather Samuel Nachod was a doctor and his grandfather was a renowned scholar, author of a Hebrew-Hungarian dictionary. Leopold's father, Samuel Weiss,
2703-903: Was a shopkeeper in Pecs. In primary schools Leopold was not doing well, so for a while his father took him away to home schooling. The future scientist developed his interest in mathematics in high school thanks to his teacher Sigismund Maksay. Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin , where he was taught by Hermann Schwarz . In 1902 he earned his doctorate from University of Budapest (today Eötvös Loránd University ). From 1902 to 1905 Fejér taught there and from 1905 until 1911 he taught at Franz Joseph University in Kolozsvár in Austria-Hungary (now Cluj-Napoca in Romania ). In 1911 Fejér
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#17327917817982756-539: Was appointed to the chair of mathematics at the University of Budapest and he held that post until his death. He was elected corresponding member (1908), member (1930) of the Hungarian Academy of Sciences . During his period in the chair at Budapest Fejér led a highly successful Hungarian school of analysis. He was the thesis advisor of mathematicians such as John von Neumann , Paul Erdős , George Pólya and Pál Turán . Thanks to Fejér, Hungary has developed
2809-529: Was not continued, and Erdős was forced to take positions as a wandering scholar at UPenn , Notre Dame , Purdue , Stanford , and Syracuse . 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 , in 1954, the Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into
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