Paul Erdös

Hungarian mathematician

• Born: March 26, 1913; Budapest, Hungary
• Died: September 20, 1996; Warsaw, Poland

Hungarian mathematician Paul Erdös published over 1,500 papers on number theory, graph theory, combinatorics, and other topics in pure mathematics. He worked with hundreds of coauthors. His contributions to mathematics include the first elementary proof of the prime number theorem, and the development of the field of probabilistic number theory.

Also known as: Pál Erdös

Primary field: Mathematics

Specialties: Number theory; combinatorics

Early Life

Paul Erdös was born on March 26, 1913, in Budapest, Hungary, into a family of Jewish descent. His parents, Anna and Lajos Erdös (Lajos had changed his family name from Engländer), were both academics who worked as high school mathematics and physics teachers. Erdös was the youngest of the family’s three children, but the only one who survived: his two older sisters died of scarlet fever around the time of his birth. In 1914, when World War I broke out and Russian forces clashed with Austro-Hungarian troops, Erdös’s father was captured as a prisoner of war and sent to Siberia. He did not return for six years. Erdös was raised by his mother and a German governess.

The death of their daughters left Erdös’s parents eager to protect their son, and he led a remarkably sheltered life. Although he attended school sporadically, Erdös received most of his education—which included training in mathematics, physics, history, and biology—at home. He also studied English, German, French, and Latin. Erdös’s mathematical talents were evident at an early age. By the time he was four years old, he was calculating the number of seconds in people’s ages, working with negative numbers, and multiplying three-digit numbers. As he grew older, he enjoyed solving advanced problems in the Journal of Mathematics and Physics for High Schools, a publication that encouraged its readers to improve on the model solutions printed within.

In 1930, Erdös enrolled in the University Pázmány Péter of Budapest, in Hungary. In Budapest, he studied under the mathematician Leopold Fejér, and developed friendships with several other young Jewish mathematicians who would become respected mathematicians in their own right, including Paul Turán and Georges Szekeres. As a student, Erdös began developing and implementing an eccentric vocabulary of terms, which he used in reference to both mathematics and everyday life. Small children, for example, he called “epsilons;” teaching mathematics was “preaching.” He labeled mathematical proofs that he found particularly pleasing as being from “The Book,” an imaginary, infinite manuscript containing the best theorems. Erdös graduated with his PhD in 1934.

Life’s Work

The thesis Erdös had completed at Pázmány Péter contained a proof for a theorem originally proposed by the nineteenth-century mathematician Joseph Bertrand. Bertrand’s postulate states that for every positive integer n that is greater than one, a prime number can be found between that number and 2n. (A prime number is a number that is evenly divisible only by itself and by the number 1.) The theorem had already been proven, but Erdös’s proof was simpler, and considered more elegant, than the existing one. It caught the eye of many established mathematicians and earned him his first serious attention from the world of mathematical research.

As his reputation grew, Erdös was granted a fellowship at the University of Manchester in England. He left home in October 1934, driven in part by a desire to escape the growing anti-Semitism in Hungary. Although he would return on visits numerous times, he never lived in the country of his birth again. Erdös spent four years in Manchester. During this time, he served as a non-teaching postdoctoral fellow with the prominent mathematician Louis Mordell and developed what would be lifelong working friendships with Harold Davenport and Richard Rado. Mordell was a number theorist; he studied the properties of numbers and the relationships between them, particularly the natural numbers (positive whole numbers). Most of the papers Erdös wrote at Manchester were about number theory.

In 1938, as the geopolitical tensions that lead to World War II were fomenting, Erdös left Europe for the United States. He accepted a temporary research appointment at Princeton University’s Institute for Advanced Study in New Jersey. The following year was one of the most productive of his life. Among other achievements, Erdös published a series of theorems with mathematicians Mark Kac and Aurel Wintner that pioneered a powerful new tool known as the probabilistic method.

Broadly speaking, the probabilistic method uses the tools of probability to determine how likely it is that a given mathematical object will have a certain property. It is applied in many areas of mathematics, including number theory and combinatorics (a branch of mathematics that treats objects as finite sets). For example, the Erdös–Kac theorem is a statement about the probability that a random integer n will have a certain number of distinct factors that are prime numbers. Today, the probabilistic method lies at the heart of discrete mathematics, which is a branch of mathematics that is used in designing and analyzing modern computer science algorithms. Erdös’s work is relevant, for example, in developing programs to encrypt information and rank web search results according to their relevance.

When Erdös’s fellowship at Princeton was not renewed, it marked the beginning of what would be nearly fifty years of perpetual wandering. He never again held a full-time position, never married, and had no permanent address. His sole purpose in life was to be free to work on mathematical problems. To that end, he would accept temporary visiting research appointments or other part-time positions at universities, living out of a suitcase in the houses of friends and colleagues. This flexibility allowed him to travel all over the United States and the world, collaborating on multiple simultaneous projects with mathematicians in France, Germany, Australia, Israel, and China, among many other countries.

Erdös was recognized with numerous awards, including the 1951 Frank Nelson Cole Prize for number theory and the 1984 Wolf Prize in Mathematics. He died of a heart attack on September 20, 1996, in Warsaw, Poland, at the age of eighty-three. He was in Poland to attend a mathematics conference.

Impact

It is difficult to define Erdös’s impact on mathematics, because the problems he was interested in were spread over so many different areas. In addition to his work on the first simple proof of the prime number theorem and his pioneering use of the probabilistic method, Erdös is remembered for his substantial contributions to the formation of Ramsey theory. Ramsey theory is a branch of combinatorics that examines how patterns or order appear in mathematical structures as they get larger. The quintessential Ramsey theory problem attempts to determine the minimum number of guests required at a party before the host can be sure that a certain number of people will be either mutual strangers or mutual acquaintances. Erdös worked on several proofs of the so-called «party guest problem,» with the guests translated into mathematical terms on a graph. He is credited with developing Ramsey theory into a fully established subfield of combinatorics.

One of the most significant ways Erdös contributed to his discipline was by expanding upon the work of other mathematicians. Because he was an itinerant who was affiliated with many institutions, Erdös was able to collaborate with hundreds of colleagues all over the world. To this day, the so-called «Erdös number» is considered a badge of honor among mathematicians: anyone who has directly coauthored a paper with Erdös can claim an Erdös number of 1, anyone who has coauthored a paper with that person can claim an Erdös Number of 2, and so on.

Erdös also inspired new mathematical research by formulating not solutions, but problems. He kept a famous list of hundreds of unsolved problems in number theory, graph theory, combinatorics, and many other fields, and offered cash rewards (which varied depending on how difficult he considered a problem to be) for anyone who offered the first good solution. Many of these open questions are still being worked through and solved by mathematicians, decades after his death.

Bibliography

Aigner, Martin, and Gunter M. Ziegler, eds. Proofs From the Book. Berlin: Springer, 2010. Print. Collection of mathematical proofs drawn from fields in which Erdös worked. Includes sections on number theory, geometry, analysis, combinatorics, and graph theory. Black and white illustrations, index.

Hoffman, Paul. The Man Who Loved Only Numbers. New York: Hyperion, 1998. Print. Includes details about Erdös’s life and career. Includes details on many prominent mathematicians who collaborated with Erdös. Source notes, index.

Schumer, Peter. “The Magician of Budapest.” in The Edge of the Universe: Celebrating 10 Years of Math Horizons. Washington, DC: Mathematical Assn. of America, 2006. 108–12. Print. Provides an introduction to Erdös. Reviews his proofs of the prime number theorem. Includes suggested reading.