Erdos, Paul

▪ 1997

      Hungarian mathematician (b. March 26, 1913, Budapest, Hung.—d. Sept. 20, 1996, Warsaw, Pol.), pioneered the fields of number theory and combinatorics and was regarded as one of the century's greatest mathematicians. At the age of 20 he discovered a proof for a classic theorem of number theory that states that there is always at least one prime number between any positive integer and its double. After receiving (1934) a Ph.D. from the University of Budapest, Erdos was awarded a postdoctoral fellowship at the University of Manchester, Eng. In 1938 he immigrated to the U.S. On fellowship at the Institute for Advanced Study in Princeton, N.J., Erdos founded the study of probabilistic number theory with Aurel Wintner and Mark Kac and proved important results in approximation theory with Paul Turan. Erdos and Atle Selberg astounded the mathematics community in 1949 by giving an elementary proof of the prime number theorem—for more than 50 years it had been assumed that no elementary proof could be given. After spending much of the 1950s in Israel, Erdos traveled almost constantly, earning a reputation as a restless "wandering scholar" who collaborated with hundreds of mathematicians on a variety of problems. "Another roof, another proof," was his legendary motto. In later years Erdos worked primarily in the field of combinatorics, an area of mathematics fundamental to computer science. When he won the Wolf Foundation Prize in 1983, he gave fellow mathematicians most of the prize money he received. He also received many honorary degrees and was a member of the Hungarian Academy of Sciences (1956), and he was a foreign associate of the academies of the U.S. (1979), India (1988), and the U.K. (1989). At the time of his death, Erdos had published more than 1,500 mathematical papers.

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▪ Hungarian mathematician

born March 26, 1913, Budapest, Hungary

died September 20, 1996, Warsaw, Poland

      Hungarian “freelance” mathematician (known for his work in number theory and combinatorics) and legendary eccentric who was arguably the most prolific mathematician of the 20th century, in terms of both the number of problems he solved and the number of problems he convinced others to tackle.

      The son of two high-school mathematics teachers, Erdös had two sisters, ages three and five, who contracted scarlet fever and died the day he was born. His mother, fearing that he, too, might contract a fatal childhood disease, kept him home from school until the age of 10. With his father confined to a Russian prisoner-of-war camp for six years and his mother working long hours, Erdös passed the time flipping through his parents' mathematics books. “I fell in love with numbers at a young age,” Erdös later recalled. “They were my friends. I could depend on them to always be there and always behave in the same way.” At three he entertained his mother's friends by multiplying three-digit numbers in his head, and at four he discovered negative numbers. “I told my mother,” he said, “that if you take 250 from 100, you get –150.”

      In 1930, at age 17, Erdös entered the Péter Pázmány University in Budapest, where in four years he completed his undergraduate work and earned a Ph.D. in mathematics. Of all the numbers, it was the primes (integers such as 2, 3, 5, 7, and 11 whose only divisors are 1 and themselves) that were Erdös's “best friends.” As a college freshman, he made a name for himself in mathematical circles with a stunningly simple proof of Chebyshev's theorem, which says that a prime can always be found between any integer (greater than 1) and its double. Even at this early point in his career, Erdös had definite ideas about mathematical elegance. He believed that God, whom he affectionately called the S.F. or Supreme Fascist, had a transfinite book (“transfinite” being a mathematical concept for something larger than infinity) that contained the shortest, most beautiful proof for every conceivable mathematical problem. The highest compliment he could pay to a colleague's work was to say, “That's straight from The Book.” As for Chebyshev's theorem, no one doubted that Erdös had found The Book proof.

      During his university years he and other young Jewish mathematicians, who called themselves the Anonymous group, championed a fledgling branch of mathematics called Ramsey theory, which has as its philosophical underpinning the idea that complete disorder is impossible. A concrete example is the random scattering of points on a plane (a flat surface). The Ramsey theorist conjectures that no matter how haphazard the scattering appears, certain patterns and configurations of points must emerge.

      In 1934 Erdös, disturbed by the rise of anti-Semitism in Hungary, left the country for a four-year postdoctoral fellowship at the University of Manchester in England. In September 1938 he emigrated to the United States, accepting a one-year appointment at the Institute for Advanced Study in Princeton, New Jersey, where he cofounded the field of probabilistic number theory. During the 1940s he wandered around the United States from one university to the next—Purdue, Stanford, Notre Dame, Johns Hopkins—spurning full-time job offers so that he would have the freedom to work with anyone at any time on any problem of his choice. Thus began half a century of nomadic existence that would make him a legend in the mathematics community. With no home, no wife, and no job to tie him down, his wanderlust took him to Israel, China, Australia, and 22 other countries (although sometimes he was turned away at the border—during the Cold War, Hungary feared he was an American spy, and the United States feared he was a communist spy). Erdös would show up—often unannounced—on the doorstep of a fellow mathematician, declare “My brain is open!” and stay as long as his colleague served up interesting mathematical challenges.

      With amphetamines to keep him going, Erdös did mathematics with a missionary zeal, often 20 hours a day, turning out some 1,500 papers, an order of magnitude higher than his most prolific colleagues produced. His enthusiasm was infectious. He turned mathematics into a social activity, encouraging his most hermetic colleagues to work together. The collective goal, he said, was to reveal the pages in the S.F.'s Book. Erdös himself published papers with 507 coauthors. In the mathematics community those 507 people gained the coveted distinction of having an “Erdös number of 1,” meaning that they wrote a paper with Erdös himself. Someone who published a paper with one of Erdös's coauthors was said to have an Erdös number of 2, and an Erdös number of 3 meant that someone wrote a paper with someone who wrote a paper with someone who worked with Erdös. Albert Einstein's Erdös number, for instance, was 2. The highest known Erdös number is 15; this excludes nonmathematicians, who all have an Erdös number of infinity.

      In 1949 Erdös had his most satisfying victory over the prime numbers when he and Atle Selberg (Selberg, Atle) gave The Book proof of the prime number theorem (which is a statement about the frequency of primes at larger and larger numbers). In 1951 John von Neumann (von Neumann, John) presented the Cole Prize to Erdös for his work in prime number theory. In 1959 Erdös attended the first International Conference on Graph Theory, a field he helped found. During the next three decades he continued to do important work in combinatorics, partition theory, set theory, number theory, and geometry—the diversity of the fields he worked in was unusual. In 1984 he won the most lucrative award in mathematics, the Wolf Prize, and used all but $720 of the $50,000 prize money to establish a scholarship in his parents' memory in Israel. He was elected to many of the world's most prestigious scientific societies, including the Hungarian Academy of Science (1956), the U.S. National Academy of Sciences (1979), and the British Royal Society (1989). Defying the conventional wisdom that mathematics was a young man's game, Erdös went on proving and conjecturing until the age of 83, succumbing to a heart attack only hours after disposing of a nettlesome problem in geometry at a conference in Warsaw.

Paul Hoffman

Additional Reading

Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth (1998), winner of the Rhône-Poulenc General Prize for Science Books, is a highly accessible and entertaining biography. Bruce Schechter, My Brain Is Open: The Mathematical Journeys of Paul Erdös (1998), is another entertaining biography. Paul Erdös, The Mathematics of Paul Erdös, 2 vol. (1996), is a comprehensive survey of his work. N Is a Number: A Portrait of Paul Erdös (1993), directed by George Csicsery, is an award-winning documentary of his life that also contains animations illustrating some of his mathematical work.

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