Paul Erdős

Paul Erdős was born on 26 March 1913, in Budapest, Austria-Hungary,[7] the only surviving child of Anna (née Wilhelm) and Lajos Erdős (né Engländer).[8][9] His two sisters, aged three and five, both died of scarlet fever a few days before he was born.[10] 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 prisoner of war during 1914–1920,[9] 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.[11]

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.[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.[13][14]

At the age of 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 the problems that appeared each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools.[15]

Erdős began studying at the University of Budapest when he was 17 after winning a national examination. At the time, admission of Jews to universities was severely restricted.[11][16] By the time he was 20, he had found a proof for Chebyshev's theorem.[16] In 1934, at the age of 21, he was awarded a doctorate in mathematics.[16] 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 Fascist-aligned regime. While there he met Godfrey Harold Hardy and Stan Ulam.[11]

Because he was Jewish, Erdős decided Hungary was dangerous and relocated to the United States in 1938.[16] 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.[16][17] 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".[11]

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.[18] 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.[18] 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.[18]

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 the project, but the invitation was withdrawn when Erdős expressed a desire to return to Hungary after the war.[11]

On 20 September 1996, at the age of 83, he had a heart attack and died while attending a conference in Warsaw.[19] 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.[19]

Erdős never married and had no children.[8] He is buried next to his mother and father in grave 17A-6-29 at Kozma Street Cemetery in Budapest.[20] For his epitaph, he suggested "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább").[21]

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".[22]

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 Princeton University's Institute for Advanced Study for the next ten years. Despite outstanding papers with Mark Kac and Aurel Wintner on probabilistic number theory, Paul Tura 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 UPenn, Notre Dame, Purdue, Stanford, and Syracuse.[23] He would not stay long in one place, instead traveling among mathematical institutions until his death.

In 1954, the Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States.[24] 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.

Counter-clockwise from left: Erdős, Fan Chung, and her husband Ronald Graham, Japan 1986

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 including American universities in his teaching and travels. Ten years later, in 1973, the 60-year-old Erdős voluntarily left Hungary.[25]

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.[26] He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1977.[27] 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.[28][29]

Erdős's problems

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 - Ernst Strauss called him "the absolute monarch of problem posers".[42] Throughout his career, Erdős would offer payments for solutions to unresolved problems.[43] 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[44] 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.[45]

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, cannot be cashed) or a cashable check from Graham.[46]

Perhaps 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 the Green–Tao theorem). The payment for the solution of the problem is currently worth US$5,000.[47]

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

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,[49] 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.[50]

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers.[51] For example, the roughly 268,000 mathematicians with a known Erdős number have a median value of 5.[52] In contrast, the median Erdős number of Fields Medalists is 3.[53] As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or less.[54][55] 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.[56]

The Erdős number was most likely first defined by Casper Goffman,[57] an analyst whose own Erdős number is 2.[58] Goffman published his observations about Erdős's prolific collaboration in a 1969 article titled "And what is your Erdős number?"[59]

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.[60] Erdős numbers have also been proposed for an infant, a horse, and several actors.[61]

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 problems" (see below). 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 Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems",[63] and Erdős drank copious quantities; this quotation is often attributed incorrectly to Erdős,[64] but Erdős himself ascribed it to Rényi.[65] 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."[66] After he won the bet, he promptly resumed his use of Ritalin and Benzedrine.[67]

He had his own idiosyncratic vocabulary; although an agnostic atheist,[68][69] 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.[70] 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).[71][72] 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 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:[67]

• Children were referred to as "epsilons" (because in mathematics, particularly 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".[73]
• 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),[67] the Soviet Union was "joedom" (after Joseph Stalin),[67] and Israel was "isreal". He claimed that Hindi was the best language because words for old age (buddha) and stupidity (buddhu) sounded the same.[74]

• List of topics named after Paul Erdős – including conjectures, numbers, prizes, and theorems
• Box-making game
• Covering system
• Dimension (graph theory)
• Even circuit theorem
• Friendship graph
• Minimum overlap problem
• Probabilistic method
• Probabilistic number theory
• The Martians (scientists)

• Bruno, Leonard C. (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. ISBN 978-0787638139. OCLC 41497065.
• Schechter, Bruce (1998). My Brain is Open: The Mathematical Journeys of Paul Erdős. New York: Simon & Schuster. ISBN 978-0-684-84635-4.
• Bollobás, Béla (December 1996). "A Life of Mathematics – Paul Erdos, 1913-1996" (PDF). Focus. Washington, D.C.: Mathematical Association of America. 16 (6): 4. Retrieved 6 May 2022.
• Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. London: Fourth Estate Ltd. ISBN 978-1-85702-811-9.

• Aigner, Martin; Ziegler, Günther (2014). Proofs from THE BOOK. Berlin; New York: Springer. doi:10.1007/978-3-662-44205-0. ISBN 978-3-662-44204-3.

Wikiquote has quotations related to Paul Erdős.

Wikimedia Commons has media related to Paul Erdős.

• Erdős's Google Scholar profile
• Searchable collection of (almost) all papers of Erdős
• O'Connor, John J.; Robertson, Edmund F., "Paul Erdős", MacTutor History of Mathematics archive, University of St Andrews
• Paul Erdős at the Mathematics Genealogy Project
• Jerry Grossman at Oakland University. The Erdös Number Project
• The Man Who Loved Only Numbers – Royal Society public lecture by Paul Hoffman (video)
• Radiolab: Numbers, with a story on Paul Erdős
• Fan Chung, "Open problems of Paul Erdős in graph theory"