One of my math students, Cheryl Mootz, already holds a graduate degree and teaches for Laurel Oaks. She aspires to teach mathematics at the secondary school level and is working on a B.S. in Math Education from Wilmington College so as to be certified. Ms. Mootz has also taken MAT 305 History of Mathematics at Northern Kentucky University and wrote an interesting term paper on the life of Paul Erdös for that course. With her permission I am reprinting her paper below.

In the early years of this century, until the Great War intervened, Hungarians were said to have an addiction to mathematics and science. This "addiction" produced men of science who would change the world. Although the average citizen in the United States does not recognize their names, a debt of gratitude is owed these Hungarians, who were arguably genius. For instance, the structure and organization of virtually all modern computers is based on a single theoretical model of computer design called the Von Neumann architecture. The name comes from the Hungarian mathematician John Von Neumann, who first proposed it in 1946. Few people outside scientific circles recognize the name Leo Szilard, the scientist who first realized that chain reactions can unleash the power of the atom, or Edward Teller, who took that knowledge further and became the father of the hydrogen bomb. Hungarians also excelled in music and the arts, as well as other lesser-known talents. One may not recognize the name of Hungarian Eric Weiss, but would be more apt to recognize his stage name, Harry Houdini.

There are as many explanations for the origin of this extraordinary talent as there are individuals of Hungarian descent who possess it. One of the more believable explanations is that Martians came to Earth at the turn of the century in search of intelligent life. Finding none, they impregnated Hungarian women and returned home. Thus the baby boom in Hungarian intelligentsia between roughly 1900 and 1920. More likely the elucidation can be found in the strong intellectual values of the middle class Jewish inhabitants and the excellent educational system of turn of the century Hungary. Whatever the impetus for this overabundance of intelligence, Hungarians of this period contributed much to the world of mathematics. One of the greatest of these contributors was Paul Erdös.

When one reads of Paul Erdös, two words invariably come up: prolific and eccentric. While "above average" mathematicians publish some 20 articles in a lifetime, Erdös wrote over 1500 papers, books, and articles, more than any mathematician in history. To say that he was prolific, which means productive, is a fair description. The March 29, 1999 edition of Time magazine, Michael D. Lemonick writes "In a profession with no shortage of oddballs, he was the strangest. Erdös had no home, no possessions, and no life aside from mathematics." The statements are true, so eccentric is probably a fair description of Paul Erdös as well. Despite his peculiarities, Paul Erdös was arguably one of the greatest mathematicians of the twentieth century. He is credited with "one of the greatest mathematical discoveries of the twentieth century...the simple equation that two heads are better than one." (Schechter, 1998, p. 14)

At the time of Erdös' childhood in Hungary, educators were revered so much that there is a street named after John Von Neumann's teacher rather than Von Neumann himself. In a country that embraced education and educators, Paul Erdös was born into a family of educators. His grandfather was a Hungarian Jew schoolteacher. His father, Lajos Englander, attended Pazmany University in Budapest, where he studied mathematics. He would become a high school teacher of mathematics and philosophy. Anti-Semitism in Europe in the first half of this century made being Jewish an invitation for discrimination, so Lajos would change his surname to the Hungarian Erdös, which means "from the woods." Paul's mother, Anna also taught high school mathematics. The two were married in 1905 and quickly had two girls, Magda and Klara. In March 1913, as Anna was hospitalized to give birth to Paul a scarlet fever epidemic raced through Budapest. When Anna brought her newborn Pal, later westernized to Paul, home, her two daughters were dead. Fear of losing Paul would cause Lajos and Anna Erdös to coddle him in such a way that he would never be able to be completely independent.

Before Paul had reached his first birthday Austria-Hungary found itself at war with Sarajevo, and Lajos Erdös was drafted into the army. He was captured in a Russian offensive and spent six years as a prisoner of war in Siberia. Anna Erdös, having lost two daughters and her husband, reacted by increasing her protectiveness over her young son. She did not allow him to attend school because she imagined it to be "full of germs," preferring to teach him at home instead. He would have no need to learn to tie his shoes until he was eleven years old, nor to butter his own bread until he had reached the age of twenty-one.

Anna would continue to teach in the public schools, so Paul would stay with a governess, whom he despised. He was motivated to learn to count at an early age so that he could keep track of the days to summer vacation when his Anyuka (Hungarian for mother) could be at home with him. Paul showed the signs of mathematical genius as a preschooler. At three, he was multiplying three and four digit numbers in his head. At the age of four, he did what had taken his predecessors in mathematics thousands of years: he discovered negative numbers. When asked what you get when you subtract 250 from 100, he told his mother, "150 below zero." The same year, he made a discovery that he called his second greatest. While shopping with his mother, he realized that though time is infinite, life is not. He realized that he would someday die, a realization that caused him a great deal of anguish all the years of his life.

Austria-Hungary lost the Great War in 1918 marking the end of the golden years of intellectualism in Hungary. An independent Hungary underwent a period of political upheaval marked by anti-Semitism. Anna Erdös' success in being promoted to principal of her school during the war years was considered a crime by the new government, so she was barred from teaching in the public schools. She would be relegated to tutoring students in her home, which made it more convenient to work with Paul. In November 1920, when he was seven years old, Paul would be reunited with his apuka (daddy) when he returned home. Lajos Erdös resumed teaching science and math at one of Budapest's top high schools.

Paul had difficulty with the discipline and restrictions of school. He exhibited signs of hyperactivity until the day he died. When an idea sparked him, Paul would jump up from his chair, run around, flail his arms about, and talk wildly of the enlightenment that had just come to him. Paul's parents educated him primarily at home although he would spend a year or two at his father's school, Szent Tsvan Gimnasium, or at the Tavaszmezo gymnasium, where he studied math and foreign language.

Lajos introduced his son to prime numbers when he was ten years old. This would be his favorite branch of mathematics, and would be the subject in which he would make his mark as a mathematician. At the age of eleven, Paul discovered the KoMal, short for Kozepiskolai Mathematikai Lapol (The Mathematical Journal for Secondary Schools), published by Laszlo Ratz, John Von Neumann's teacher. In addition to articles by mathematicians and other educators, the KoMal contained mathematical contests. Young mathematicians such as Erdös would send their solutions back to KoMal where a panel of judges would grade them, and the best were published. Erdös's first solution was published in 1926, when he was thirteen years old. Coincidentally, Paul Turan, who would become one of Erdös's most important collaborators, solved the same problem.

Despite a six-percent cap on the number of Jewish students admitted to the postsecondary schools of Hungary, Paul was able to enter the Science University of Budapest. While a student there, he would meet his future collaborators, George Szekeres, Esther Klein, Paul Turan, and Andrew Vazsonyi. In their youth they were united in that each had solved problems for the KoMal, while they would be united in their flight from German persecution as young adults.

Since his father introduced him to prime numbers, Erdös had been intrigued by number theory, the branch of mathematics concerned with the properties of integers. In 1931, at the age of 18, Paul was able to come up with a new proof for a nearly 100 year old conjecture. In 1849, Joseph Bertrand hypothesized that there is always at least one prime number between a number and its double. Chebychev created a long and tedious proof in 1850. Erdös's more elementary proof led to new applications of Bertrand's hypothesis. Erdös wrote his Ph. D. dissertation on this as second year undergraduate.

While a student at the university, Paul Erdös also made his first contribution to Combinatorics, the branch of mathematics concerned with counting things. While it has applications in almost every branch of science and technology, it is especially important to the design of communications networks and computers. Esther Klein introduced Erdös to this field of mathematics. Esther found that when she drew five points randomly on a page with no three collinear, four of them would always define the vertices of a convex quadrilateral. She took the problem to her friends George Szekeres and Paul Erdös for assistance in writing a proof of this conjecture, which they quickly did. This would become the basis for the first of many joint papers between Erdös and Szekeres.

In generalizing Esther's problem to other numbers of random points, Szekeres and Erdös surmised that a convex polygon of n sides would always arise if 1 + 2^(n - 2) or more points were sprinkled across the plane. In their attempts to prove this generalization, Erdös and Szekeres revived a piece of mathematics that had been filed away as impossible. Erdös coined the name "Ramsey Theory" for this branch of mathematics that proves that complete disorder is impossible. Erdös liked to explain mathematical concepts in terms that anyone with a minimal amount of high school algebra could understand. His method of explanation of Ramsey Theory is known as the party problem. "What is the smallest party in which it is guaranteed that a specified number of guests will either all know each other or all be mutual strangers?" (Hoffman, 1998, p. 52) Erdös proved that in the case where the specified number is three, there must be six guests. If you want a foursome, you must invite eighteen guests. Beyond that, no mathematician has yet been able to prove the exact numbers necessary without the aid of a computer. Erdös is credited with opening up this entire new field of mathematics.

In 1934, at the age of 21, Erdös received his doctoral degree from Pazmany University. He was one of the youngest persons ever to do so. He was offered a fellowship at the University of Manchester, where he would do post-doctoral work until the late 1930's. As Jewish oppression increased in pre-World War II Hungary, it became apparent that he could not return to his homeland, so Paul prepared to flee to the United States.

In October 1938, Paul arrived at the Princeton Institute for Advanced Study, where he had been awarded a stipend for the 1938-1939 academic year. The Institute for Advanced Study was financed by the Bamberger siblings who had amassed a fortune in the retail business, and managed to keep it by selling out just prior to the stock market crash of 1929. Albert Einstein was signed on as the Institute's first professor, further making it a "prized destination for the world's intellectual elite." (Schechter, 1998, p. 102)

While at the Institute, Paul began to produce mathematical papers in massive quantities. Of his first sixty papers, all but two were on the topic of number theory. At that time, graph theory and its close kin, combinatorics, were considered undesirable areas of mathematics. In 1938, Paul Turan wrote a paper that became the basis of a new field of study called extremal graph theory, which combined number theory and graph theory, making these two fields more attractive and acceptable to mathematicians. Erdös became one of the leading contributors of research to this important new field. His first paper on the subject was published shortly after he started research at the Institute. He also became interested in Probability theory. He is credited with the creating the Probabilistic Method, otherwise known as the Erdös Method. With Mark Kac, Erdös created a theorem joining normal law and number theory, creating yet another new branch of mathematics, probabilistic number theory. The Erdös-Kac theorem states that the number of distinct prime factors of the integers less than N follow the normal distribution curve.

After one year, Erdös was non-renewed at the Institute for Advanced Study. Lack of funding was the reason given, but there are those who believe that Erdös's collaboration was threatening to the great minds who studied there. Nevertheless, Oswald Veblen, the head of the Institute's math department, was able to secure $750 for him, enough to support Erdös for the second term of the 1939-1940 academic year.

World War II and its aftermath would create problems for Erdös. During those years Erdös survived on fellowships of one year or less duration, lecture fees, and the kindness of his colleagues. His first year after leaving the Institute was spent as a Harrison Fellow at the University of Pennsylvania. That year, 1941, was not a very productive year for Erdös, probably due to the depression he was feeling over the war, and worry about the fate of his Jewish family and friends back home in Hungary. In that same year, he was suspected of spying when he and two mathematician friends, one British and one Japanese, wound up in the wrong place at the wrong time. The trio had gone sight-seeing in New York and unwittingly ended up taking photographs of each other in front of a secret radar installation.

One of the papers that Erdös wrote in 1941 was the result of collaboration with the Polish mathematician Stanislaw Ulam at the University of Wisconsin. In 1942, Ulam went to work with John Von Neumann on the Manhattan Project, the making of the atomic bomb. Ulam suggested that Erdös be invited to work on the project. Erdös, however, when approached with the prospect by Edward Teller, made the mistake of telling him that he might want to return to Hungary after the war. Hungary was allied with Germany, so Erdös was considered a security threat and was not allowed to work on the project.

However, one of Erdös's conjectures is credited with saving Western civilization. By the World War II era, Erdös had gotten into the habit of offering rewards for solutions to problems he was unable to solve. The amount of the award would vary by Erdös's perception of the difficulty of the problem. William Tutte, a British chemist worked with his two college roommates (mathematicians) on solving one of these problems. Working on this problem piqued Tutte's interest in mathematics, so when he did not find success in his chosen field he became a mathematician. He would eventually help crack a secret German communication code, enabling the allies to intercept and destroy German supply ships in World War II.

In 1942, Paul received word that his father had died of a heart attack at the age of 63. By the time the German army had occupied Hungary in March of 1944, communication with his home country had stopped. Not until the war was over in 1945 would Paul learn that his mother had survived the Holocaust, but many relatives and friends, including the editor of the KoMal had been exterminated.

In the years after the war, Paul collaborated with a multitude of mathematicians and produced a vast amount of original mathematics in such areas as probability theory, combinatorics, graph theory, geometry, interpolation, and number theory. In 1947, an incident would occur that should have brought him great joy and prestige, but instead caused controversy and conflict. The incident is what some claim to be the reason for Paul Erdös's nearly fifty years of wandering the globe.

In 1896, Hadamard and Poussin proved the Prime Number Theorem. The proof is not an elementary proof because it is hard to relate to the properties of integers. It had been Erdös's lifelong dream to create an elementary proof of the Prime Number Theorem, which explains the pattern of distribution of prime numbers. In 1947, a Norwegian mathematician had been brought to the Institute for Advanced Study by professor Hermann Weyl. Selberg had derived a proof of the Dirichlet Theorem, dealing with the occurrence of primes in arithmetic progressions. In proving this, he had created what he called the "fundamental formula" which eventually proved to be the key to an elementary proof of the Prime Number Theorem. Selberg showed his notes on the fundamental formula to Erdös's friend Paul Turan. Turan, not knowing the significance of the formula, shared this with Erdös and others at the Institute during a lecture on Selberg's proof of the Dirichlet Theorem. Brilliant as he was, Erdös realized immediately that the fundamental formula would help prove the Prime Number Theorem. He went immediately to Selberg in hopes of collaborating on a proof: Selberg was not the social mathematician that Erdös was, preferring to work alone and take all credit for his discoveries, so he tried to throw Erdös off track by telling him that the proof could not be done with the fundamental formula. Erdös was not sidetracked and was able to create an elementary proof of the Prime Number Theorem. Selberg would not agree to a joint publication, but did agree to publish his paper in one journal while Erdös published his in another, each giving the other credit for the work he had done. Selberg became annoyed when he heard his colleagues giving Erdös credit for the proof, and published his first. Erdös had difficulty getting his proof published, supposedly because Weyl cast doubt on Erdös's right to publish. In 1950, Selberg won the Fields Medal for this proof. Erdös was honored in 1952 with the Cole Prize for his work on the proof.

By the time this incident took place, Erdös had spent over a decade in the United States. Despite his lengthy stay in the U.S., he had never bothered to update his student visa. The Cold War caused great tension between the U.S., which Erdös called "Sam", and the Soviet Union, which Erdös called "Joe." To make matters worse, the communist witch hunting of McCarthyism was in full swing. Travel abroad was difficult during the post-war McCarthyism era, and Erdös's immigration status limited his traveling, thus his collaboration with other mathematicians. In 1948 he was able to obtain a green card that would allow him reentry to the U.S. after a trip to Europe and Hungary. In 1949, the political climate in Hungary made it painfully obvious that he could not make Hungary his permanent home, nor could he visit, lest he not be permitted to leave. In the late forties and early fifties, Erdös divided his time between the United States and England.

In 1953, Erdös was working at the University of Notre Dame in South Bend, Indiana. Erdös was an atheist, referring to God as the "Supreme Fascist," or "S.F." He believed that the S.F. had a book of all the most elementary proofs and he denoted a particularly elegant proof as "one from the book." Consequently, he took a lot of ribbing from his coworkers at the Roman Catholic university. Although offered a continuing contract, Erdös turned it down, preferring instead to continue his life of traveling. Shortly thereafter, in 1954, Erdös left the U.S. to attend the International Congress of Mathematicians in Amsterdam without a reentry visa. The reentry visa was denied on the basis of an interview with an INS agent prior to his departure. In that interview Erdös refused to denounce his communist colleagues, and when asked what he thought of Karl Marx, Erdös answered, "I am not competent to judge. But no doubt he was a great man." (Schechter, 1998, p. 164)

Erdös found refuge and reluctantly gained citizenship in Israel after the conference, having been unable to do so in Holland or England. He became a "permanent visiting professor" at the Technion in Haifa (Schechter, 1998, p. 165). Between the years of 1954 and 1963, Erdös was allowed back in the United States for one brief stay to attend a mathematics conference in Boulder, Colorado in 1959. During this time, he visited Hungary frequently and was elected to the Hungarian Academy of Sciences. He was a frequent traveler, happily collaborating with old friends and new talent he had discovered.

When Erdös began writing mathematical papers, he did so as a solo act, for this was the modus operandi in the mathematical world. In 1940, individuals wrote 90 percent of all mathematical papers. After Erdös's influence, today only about half of such papers is written independently. It would be impossible to outline the contents of the over 1500 publications credited at least partially to Erdös, or to mention his nearly 500 collaborators. His efforts have been described mathematically by the quantity called the Erdös number. Mathematicians assign Erdös the number zero. As of March 30, 1999, there were 492 people who had co-authored a paper with Erdös, thus earning the Erdös number one. The 5,608 people who wrote a paper with someone who co-authored with Erdös are identified with an Erdös number of two. A person who has written a joint mathematical paper with a person having an Erdös number n earns the Erdös number n + 1. Anyone who has not collaborated with someone who has an Erdös number is assigned the Erdös number infinity. Graham (1999) notes, "All Fields Medal winners through 1998 have Erdös numbers less than 6 and...at least 63 Nobel prize winners have Erdös numbers less than 9."

When he was not traveling, Paul lived with his mother for the duration of his exile from the United States. She handled all the details of his life so he could concentrate on mathematics. She handled his finances, kept his papers filed, and sent reprints when requested. Erdös was permitted to return to the United States in 1964 and never experienced problems with his visa again, so extended his travels to the U.S. once again. Anyuka missed Paul so much when he traveled that she began to accompany him in 1964, when she was age 84. Some references assert that she would not allow him to sleep anywhere but in the same room as she.

After his mother's death in 1971, Paul Erdös's eccentricities began to become more and more evident. He refused to enter the Budapest apartment they had shared, so he would infringe upon his colleagues for a place to stay when he was in the area. He would call in the middle of the night, disregarding time zones. He'd show up on a colleague's doorstep and announce, "My brain is open!" then proceed to work mathematical problems with whoever was willing and had the energy. He began to work nineteen hours a day with the help of amphetamines and strong coffee.

Ronald Graham, an AT&T mathematician who had met Erdös at the Boulder conference in 1964, took on the task of handling Erdös's affairs after Anna Erdös's death. In 1979, Graham bet Erdös $500 that he couldn't quit taking the amphetamines for a month. Just to prove that he wasn't addicted, Erdös quit for a month, collected his $500, and then promptly began taking them again. He claimed that he couldn't get any work done without them, and scolded Graham for setting back mathematical progress by one month. Erdös had a longstanding habit of offering cash awards for the solutions to problems he was working on. The rewards ranged from $1 for a simple problem to $10,000 for one he considered hopeless. It was a good thing that the solvers of these problems coveted Erdös's signature, because the prizes were usually paid with checks from long closed accounts. Eventually Graham would have to force Erdös to start depositing his lecture money into a genuine account because the solvers began to want both the cash and the cancelled check for framing. Only a handful of Erdös's problems were ever solved in his lifetime, so he paid out less than $5,000 in rewards. His friends, including Graham have vowed to make good on the rewards if any of Erdös's problems are solved in the future.

Money was not important to Paul Erdös. In 1984, he won the Wolf Prize, one of the highest forms of recognition in mathematics. The prize carries with it a cash award of $50,000. Erdös kept only $720 of the award. Most of the money was given in gratitude to the Technion in Israel. A $30,000 endowment was given to fund a post doctoral fellowship in the name of his parents. The remainder was given to relatives, graduate students, and colleagues.

Although it was not the only honorary degree he ever received, Erdös was given an honorary degree from the University of Georgia in 1995. After the ceremony Erdös was asked to sign a baseball, which had just been autographed by Hank Aaron, who also received an honorary degree that day. Some now claim that Hank Aaron has an Erdös number of one.

Paul Erdös continued to travel and do original mathematics until the day he died. He refused to get an operation to remove the cataracts from his eyes because it would impede his work. He had a heart condition that required the insertion of a pacemaker, an operation that usually requires a twenty-four hour hospital stay. Erdös refused the overnight stay, as it would cause him to miss part of a mathematics conference. Instead, he had the operation done and took his two cardiologists with him to the conference that same day.

Erdös has been referred to as the Johnny Appleseed of Mathematics. In his final years, Erdös had become more forgetful and somewhat slower, but he continued to travel the world, spreading his love of mathematics as Johnny spread seeds. According to Erdös, to "die" was to quit doing mathematics. To "leave" was physical death. In Warsaw, Poland for a combinatorics meeting, Erdös "left" at the age of 83, of a heart attack.

Grossman, J. (1999, March 28). The Erdös Number Project. [Homepage of the Erdös Number Project], [Online]. Available: http://www.acs. oakland.edu/~grossman/erdoshp.html

Hoffman, P. (1998). The Man Who Loved Only Numbers. New York: Hyperion.

Lemonick, M. (1999, March 29). Paul Erdös: The Oddball's Oddball. Time, p. 134.

MacTutor History of Math [Homepage of Paul Erdös], [Online]. (1996, October). Available: http://history.math.csusb.edu/Mathematicians/Erdos.html [1999, March 14].

Paul Erdös [Homepage of University of Southern Denmark], [Online]. (1996, November 12). Available: http://www.imada.ou.ak/~btoft/erdoes.html [1999, March 14].

Peterson, I. (1996). Ivars Peterson's MathLand. [Homepage of Ivars Peterson], [Online]. Available: http://www.maa.org/mathland/html [1999. March 14].

Schechter, B. (1998). My Brain is Open. New York: Simon and Schuster.