# Paul Erdos Facts

**For Paul Erdos (1913-1996), mathematics was life. Number theory, combinatorics (a branch of mathematics concerning the arrangement of finite sets), and discrete mathematics were his consuming passions. Everything else was of no interest: property, money, clothes, intimate relationships, social pleasantries—all were looked on as encumbrances to his mathematical pursuits.**

A genius in the true sense of the word, Erdos traveled the world, living out of a suitcase, to problem solve—and problem pose—with his mathematical peers. A small, hyperactive man, he would arrive at a university or research center confident of his welcome. While he was their guest, it was a host's task to lodge him, feed him, do his laundry, make sure he caught his plane to the next meeting, and sometimes even do his income taxes. Cosseted by his mother and by household servants, he was not brought up to fend for himself. Gina Bari Kolata, writing in Science magazine, reports that Erdos said he "never even buttered his own bread until he was 21 years old."

Yet this man, whom Paul Hoffman called "probably the most eccentric mathematician in the world" in the Atlantic Monthly, more than repaid his colleagues' care of him by giving them a wealth of new and challenging problems—and brilliant methods for solving them. Erdos laid the foundation of computer science by establishing the field of discrete mathematics. A number theorist from the beginning, he was just 20 years old when he discovered a proof for Chebyshev's theorem, which says that for each integer greater than one, there is always at least one prime number between it and its double.

Erdos was born in Budapest, Hungary, on March 26, 1913. His parents, Lajos and Anna Erdos, were high school mathematics teachers. His two older sisters died of scarlet fever when he was an infant, leaving him an only child with a very protective mother. Erdos was educated at home by his parents and a governess, and his gift for mathematics was recognized at an early age. It is said that Erdos could multiply three-digit numbers in his head at the age three, and discovered the concept of negative numbers when he was four. He received his higher education from the University of Budapest, entering at the age of 17 and graduating four years later with a Ph.D. in mathematics. He completed a postdoctoral fellowship in Manchester, England, leaving Hungary in the midst of political unrest in 1934. As a Jew, Hungary was then a dangerous place for him to be. During the ensuing Nazi era, four of Erdos's relatives were murdered, and his father died of a heart attack in 1942.

In 1938, Erdos came to the United States. However, because of the political situation in Hungary, he had difficulty receiving permission from the U. S. government to come and go freely between America and Europe. He settled in Israel and did not return to the United States until the 1960s. While in the U.S., he attended mathematical conferences, met with top mathematicians such as Ronald Graham, Ernst Straus and Stanislaw Ulam, and lectured at prestigious universities. His appearances were irregular, owing to the fact that he had no formal arrangements with any of the schools he visited. He would come for a few months, receive payment for his work, and move on. He was known to fly to as many as fifteen places in one month—remarking that he was unaffected by jet lag. Because he never renounced his Hungarian citizenship, he was able to receive a small salary from the Hungarian Academy of Sciences.

## An Erdos Number Conveyed Prestige

So esteemed was Erdos by his colleagues that they invented the term "Erdos number" to describe their close connections with him. For example, if someone had coauthored a paper with Erdos, they were said to have an Erdos number of one. If someone had worked with another who had worked with Erdos, their Erdos number was two, and so on. According to his obituary in the New York Times, 458 persons had an Erdos number of one; an additional 4,500 could claim an Erdos number of two. It is said that Albert Einstein had an Erdos number of two. Ronald Graham, director of information sciences at AT and T Laboratories, once said that research was done to determine the highest Erdos number, which was thought to be 12. As Graham recalled, "It's hard to get a large Erdos number, because you keep coming back to Erdos." This "claim to fame" exercise underscores Erdos's monumental publishing output of more than 1,500 papers, and is not only a tribute to his genius but also to his widespread mathematical network.

Throughout his career, Erdos sought out younger mathematicians, encouraging them to work on problems he had not solved. He created an awards system as an incentive, paying amounts from $10 to $3,000 for solutions. He also established prizes in Hungary and Israel to recognize outstanding young mathematicians. In 1983, Erdos was awarded the renowned Wolf Prize in Mathematics. Much of the $50,000 prize money he received endowed scholarships made in the name of his parents. He also helped to establish an endowed lectureship, called the Turan Memorial Lectureship, in Hungary.

## Perfect Proofs from God's "Great Book"

Erdos's mathematical interests were vast and varied, although his great love remained number theory. He was fascinated with solving problems that looked—but were not—deceptively simple. Difficult problems involving number relationships were Erdos's special forte. He was convinced that discovery, not invention, was the way to mathematical truth. He often spoke in jest of "God's Great Mathematics Book in the Sky," which contained the proofs to all mathematical problems. Hoffman in the Atlantic Monthly says "The strongest compliment Erdos can give to a colleague's work is to say, 'It's straight from the Book."'

## Mother's Death Brought on Depression

Erdos's mother was an important figure in his life. When she was 84 years old, she began traveling with him, even though she disliked traveling and did not speak English. When she died of complications from a bleeding ulcer in 1971, Erdos became extremely depressed and began taking amphetamines. This habit would continue for many years, and some of his extreme actions and his hyperactivity were attributed to his addiction. Graham and others worried about his habit and prevailed upon him to quit, apparently with little result. Even though Erdos would say, "there is plenty of time to rest in the grave," he often talked about death. In the eccentric and personal language he liked to use, God was known as S.F. (Supreme Fascist). His idea of the perfect death was to "fall over dead" during a lecture on mathematics.

Erdos's "perfect death" almost happened. He died of a heart attack in Warsaw, Poland, on September 20, 1996, while attending a mathematics meeting. As news of his death began to reach the world's mathematicians, the accolades began. Ronald Graham, who had assumed a primary role in looking after Erdos after his mother's death, said he received many electronic-mail messages from all over the world saying, "Tell me it isn't so." Erdos's colleagues considered him one of the 20th century's greatest mathematicians. Ulam remarked that it was said "You are not a real mathematician if you don't know Paul Erdos." Straus, who had worked with Einstein as well as Erdos, called him "the prince of problem solvers and the absolute monarch of problem posers," and compared him with the great 18th-century mathematician Leonhard Euler. Graham remarked, "He died with his boots on, in hand-to-hand combat with one more problem. It was the way he wanted to go."

## Books

Mathematical People, Profiles and Interviews. Edited by Donald J. Albers and G.L. Alexanderson. Contemporary Books, Inc., 1985.

Ulam, S. M. Adventures of a Mathematician. Charles Scribner's Sons, 1976.

## Periodicals

The Atlantic Monthly, November 1987.

The New York Times. September 24, 1996.

Science, April 8, 1977.

Two-Year College Mathematics Journal, 10, 1979.

## Online

"In Memoriam: Paul Erdos." February 11, 1997. http://www.cs.uchicago.edu/groups/theory/erdos.html (July 20, 1997).