Paul Erdős

• 1. Introduction

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  Paul Erdős, the Hungarian mathematician born in 1913, spent most of his adult life living out of a suitcase. Time magazine called him "The Oddball's Oddball". His biographer Paul Hoffman went further and described him as "probably the most eccentric mathematician in the world". Yet the oddity had a single purpose. Erdős organized his days around mathematics and around the company of other mathematicians, and he devoted nearly every waking hour to both. Three questions run through this documentary. What kind of childhood produces a person who gives up a fixed home for the sake of unsolved problems? Why do scientists in fields far from his own still measure their distance from him? And what did he hope his own ending would look like?

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• 2. A Budapest Childhood

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  On the 26th of March 1913, Paul Erdős was born in Budapest, then part of Austria-Hungary. His parents, Anna and Lajos, were both Jewish and both taught high school mathematics. Two of their daughters, aged three and five, had died of scarlet fever only days before his birth. He became the only surviving child, and he grew especially close to his mother. Reportedly, the two shared a bed until he left for college.

  From 1914 to 1920, his father was held in Siberia as an Austro-Hungarian prisoner of war. His mother worked long hours to keep the household going, so a German governess helped raise Paul. In captivity, Lajos taught himself English but mispronounced many words. When he later taught his son the language, Paul copied those pronunciations and kept them for the rest of his life.

  At the age of five, Paul could take anyone's age and calculate in his head how many seconds that person had lived. He taught himself to read from the mathematics books his parents left around the house. When he was 16, his father introduced him to infinite series and set theory, two subjects he loved for life. In high school he became an ardent solver of the monthly problems in KöMaL, the "Mathematical and Physical Journal for Secondary Schools".

  At 17, Erdős won a national examination and began studying at the University of Budapest. Jewish admission to Hungarian universities was then severely restricted under a rule called the numerus clausus. In 1933, he joined a group of students who met often at the Anonymous statue in City Park to talk mathematics. Among them were George Szekeres, Esther Klein, her lifelong friend Márta Wachsberger, and George Svéd.

  Klein proposed a problem and offered a proof. Szekeres and Erdős then wrote a paper in 1935 that generalised the result. Erdős named the original puzzle the "Happy ending problem", because it ended in the marriage of George and Esther Szekeres.

  By the time he was 20, Erdős had proved Bertrand's postulate, and his proof was far neater than the original by Chebyshev. In 1934, at 21, he earned his doctorate in mathematics. His thesis advisor was Lipót Fejér, who also advised John von Neumann, George Pólya and Pál Turán.

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• 3. Life on the Road

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  In Manchester, England, Erdős met G. H. Hardy and Stanisław Ulam while holding a post-doctoral fellowship at Victoria University of Manchester. Because he was Jewish, Hungary had become dangerous, and in 1938 he moved to the United States. His first American position was a scholarship at the Institute for Advanced Study in Princeton, New Jersey. He wrote outstanding papers there with Mark Kac and Aurel Wintner on probabilistic number theory, with Pál Turán on approximation theory, and with Witold Hurewicz on dimension theory.

  The fellowship was not continued. Princeton had found him "uncouth and unconventional", and it extended his stay by only six months instead of the expected year. Erdős became a wandering scholar, taking positions at the University of Pennsylvania, Notre Dame, Purdue, Stanford and Syracuse. He never stayed long in one place again and traveled among mathematical institutions until his death.

  In 1941, police arrested Erdős and two others for looking around a secret radio tower. He reportedly explained that he had missed the warning signs because "I was thinking about mathematical theorems." The FBI kept a file on him and watched him into the 1970s, but found nothing incriminating.

  In 1943, Erdős began a part-time appointment at Purdue University. That same year Ulam invited him to join the Manhattan Project at Los Alamos, New Mexico, alongside other mathematicians and physicists. Erdős answered that he wished to return to Hungary after the war.

  Near the end of 1948, he finally visited Hungary and was reunited with his surviving family and friends. For the next three years he moved back and forth between England and the United States. In 1952 he accepted a temporary post at the University of Notre Dame, which let him travel for joint research whenever he liked. Despite encouragement from the university and his friends, he refused to make it permanent.

  In 1954, the Red Scare and McCarthyism caught up with him. The United States Immigration and Naturalization Service denied Erdős a re-entry visa. Officials cited his correspondence with a Chinese mathematician who had returned from the United States to China, and his 1941 FBI record. Hungary, then part of the Warsaw Pact, limited its own citizens' travel. Yet in 1956 it gave Erdős the exclusive privilege of entering and leaving as he pleased.

  During this period he spent around ten years in Israel. The Hebrew University of Jerusalem gave him a three-month position, and the Technion in Haifa then made him a "permanent visiting professor". He asked the American authorities to reconsider at intervals during the early 1960s, and a visa was finally granted in November 1963. He resumed teaching at American institutions and traveling to them.

  In 1973, the 60-year-old Erdős voluntarily left Hungary. In 1985 his longtime friend and collaborator George Szekeres brought him to Adelaide, South Australia, where he visited Flinders University and the University of Adelaide. At the latter he met a budding mathematician named Terence Tao, then 10 years old. Erdős reportedly enjoyed working with children.

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• 4. Collaboration and the Erdős Number

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  Around 1,525 mathematical articles carry Erdős's name, a total that remained unsurpassed as of 2023. Admirers compared his sheer output with that of Euler. Most of the papers had co-authors, and over his lifetime he worked with 511 different collaborators.

  András Sárközy led the list with 62 joint papers, followed by András Hajnal with 56 and the American Ralph Faudree with 50. Richard Schelp wrote 42 papers with him. Cecil C. Rousseau and Vera T. Sós each wrote 35, and Alfréd Rényi wrote 32. Pál Turán, Endre Szemerédi and Ron Graham appear further down, with 30, 29 and 28 papers.

  His doctoral students included Joseph Kruskal, George B. Purdy, Alexander Soifer, Béla Bollobás and Géza Fodor.

  In 1969, Casper Goffman published an article titled "And what is your Erdős number?" Goffman, an analyst, most likely defined the idea first. His own number is 2, because he co-authored with Richard B. Darst, who co-authored with Erdős.

  Friends of Erdős adopted the number as a tribute to his output. Erdős alone holds the number 0. His direct co-authors can claim 1, their collaborators have a number of at most 2, and so on outward. The American Mathematical Society offers a free online tool that finds the number for every author in the Mathematical Reviews catalogue.

  Some have estimated that 90 percent of the world's active mathematicians have a number smaller than 8, which fits the small-world phenomenon. Among roughly 268,000 mathematicians with a known number, the median is 5. For Fields Medalists, the median is 3. As of 2015, approximately 11,000 mathematicians had a number of 2 or lower. Distances will grow over long time scales, because mathematicians with low numbers die and can no longer collaborate.

  Many scientists in physics, engineering, biology and economics also have Erdős numbers, thanks to collaborations with mathematicians. Jerry Grossman has argued that the Baseball Hall of Famer Hank Aaron could count as a number 1. Both men autographed the same baseball for Carl Pomerance on the day Emory University awarded them honorary degrees. Numbers have even been proposed for an infant, a horse and several actors.

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• 5. The Problem Solver

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  Timothy Gowers wrote an essay, "The Two Cultures of Mathematics", that contrasts problem solvers with theory developers. Erdős belonged firmly to the first group. He pursued open questions in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory and probability theory. Most of his work centered on discrete mathematics, where he cracked many unsolved problems.

  Joel Spencer observed that Erdős's place among 20th-century mathematicians "is a matter of some controversy". The reason, in Spencer's words, is that he "resolutely concentrated on particular theorems and conjectures throughout his illustrious career". Erdős never won the Fields Medal, which was the highest mathematical prize available in his lifetime.

  Two contributions stand out above the rest. One is the development of Ramsey theory, which studies the conditions under which order must appear. The other is the application of the probabilistic method. Extremal combinatorics also owes him a whole approach, which grew partly out of the tradition of analytic number theory.

  With Atle Selberg, Erdős found the first elementary proof of the prime number theorem. The circumstances leading up to the proofs, along with disagreements over publication, produced a bitter dispute between the two men.

  Topology held little real interest for Erdős, yet he left a mark there too. He is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional. That example is now called the Erdős space.

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• 6. Problems With Price Tags

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  Ernst Strauss called Erdős "the absolute monarch of problem posers". Erdős attached cash rewards to unsolved problems. A prize of $25 marked a question he judged just out of reach of current thinking, his own included. At the top, $10,000 went on problems that were both hard to attack and mathematically significant. The most lucrative one, his conjecture on prime gaps, was solved in 2014, and the $10,000 was paid.

  Perhaps the most notable open prize concerns arithmetic progressions. The conjecture says that if the sum of the reciprocals of a sequence of integers diverges, the sequence contains arithmetic progressions of arbitrary length. A proof would settle several other open problems in number theory. One main consequence, that the prime numbers contain arbitrarily long arithmetic progressions, has already been proved independently as the Green-Tao theorem. The prize for the full conjecture is currently worth US$5,000.

  The Collatz conjecture, also called the 3N + 1 problem, is likely the most familiar problem with an Erdős prize. He offered $500 for a solution.

  At least a thousand problems are thought to remain unsolved, though no official or comprehensive list exists. The offers stayed active after Erdős died. Ronald Graham acted as the informal administrator of solutions. A solver could receive either an original check signed by Erdős, kept only as a memento because it cannot be cashed, or a cashable check from Graham. The Combinatorics Foundation, currently chaired by Steve Butler, later took over that role. In 2024, the British mathematician Thomas Bloom started a website dedicated to the problems.

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• 7. Habits and Vocabulary

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  During the Second World War, Erdős's father, two aunts and two uncles were among the relatives murdered in Budapest in the Holocaust. His mother was the only one in the family who survived. Erdős was living in America at the time. He never married and had no children.

  Alfréd Rényi coined the line "A mathematician is a machine for turning coffee into theorems". It is often credited to Erdős by mistake, though Erdős himself ascribed it to Rényi, and he drank copious quantities of coffee. After his mother died in 1971, he started taking antidepressants and amphetamines. His friends worried, and Ron Graham bet him $500 that he could not stop for a month.

  Erdős won the bet but complained about the cost. "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." Once the bet ended, he promptly resumed his use of Ritalin and Benzedrine.

  An agnostic atheist, Erdős still spoke of "The Book", an imagined volume in which God had written the most elegant proof of every theorem. He used the phrase since at least the late 1970s. In a 1985 lecture he said, "You don't have to believe in God, but you should believe in The Book." He nicknamed God the SF, short for "Supreme Fascist", and accused him of hiding socks and Hungarian passports and of keeping the best proofs to himself. A beautiful proof drew the exclamation "This one's from The Book!", which later inspired a book titled Proofs from the Book.

  His private vocabulary ran much further. Children were "epsilons", after the Greek letter mathematicians use for an arbitrarily small quantity. Women were "bosses" who "captured" men as "slaves" by marrying them, and divorced men were "liberated". Mathematicians who stopped doing mathematics had "died", while people who actually died had "left". Alcoholic drinks were "poison", and any music other than classical was "noise". A hack was a "Newton".

  Giving a lecture meant "to preach", and lectures were "sermons". Giving an oral exam meant "to torture" students. The United States became "samland", after Uncle Sam, and the Soviet Union became "joedom", after Joseph Stalin. He also claimed Hindi was the best language, since its words for old age and stupidity, buddhā and buddhū, sound almost the same.

  The Hungarian letter ő in his surname is often replaced by Erdos or Erdös, either "by mistake or out of typographical necessity". He signed his own name "Paul Erdos P.G.O.M.", short for "Poor Great Old Man". At 60 he added "L.D." for "Living Dead", and at 65 he added "A.D." for "Archaeological Discovery". At 70 came a second "L.D.", meaning "Legally Dead". At 75 he added "C.D.", which stood for "Counts Dead".

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• 8. Honours and Farewell

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  The Wolf Prize of 1983 was worth $50,000, but Erdős needed little money. He gave away most of what he earned lecturing at conferences, passing it to students or using it as prizes for problems he had posed. The prize citation praised him for "personally stimulating mathematicians the world over".

  In the last decades of his life, he received at least 15 honorary doctorates. He joined the scientific academies of eight countries, including the United States National Academy of Sciences and the Royal Society in the United Kingdom. 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 the University of Waterloo over what he considered unfair treatment of his colleague Adrian Bondy.

  Other honours included the AMS Cole Prize in Number Theory in 1951 and a plenary talk at the British Mathematical Colloquium in 1957. The London Mathematical Society made him an honorary member in 1973, and he spoke at the International Congresses of Mathematicians in 1983.

  On the 20th of September 1996, at the age of 83, Erdős had a heart attack and died while attending a conference in Warsaw. The circumstances came close to what he had wished for. He once said he wanted to be finishing an important proof at the blackboard when someone shouted, "What about the general case?" He would smile, answer "I'll leave that to the next generation," and then "keel over".

  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." In Hungarian, that reads "Végre nem butulok tovább".

  The World Federation of National Mathematics Competitions established the Paul Erdős Award in 1996, and it continues today. Two biographies appeared in 1998, Hoffman's The Man Who Loved Only Numbers and Schechter's My Brain is Open. Deborah Heiligman wrote a children's picture book in 2013, The Boy Who Loved Math: The Improbable Life of Paul Erdős. George Csicsery had already made a documentary in 1993, N is a Number: A Portrait of Paul Erdős, while Erdős was still alive.

  In 2019, the Australian author Miriam Sved, granddaughter of Márta and George Svéd, published the novel A Universe of Sufficient Size. It follows Jewish mathematicians in Hungary who flee the country, and several characters are loosely based on Esther and George Szekeres and on Erdős. In 2021, the minor planet 405571 was named "Erdőspál" in his memory. In 2025, the arachnologists Danniella Sherwood and R. C. Gallon described a new tarantula from Nigeria, Heterothele erdosi, named in his honour.

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