Some links to internet pages with additional information on Paul Erdos:

http://www.math-inst.hu/staff/erdos.html

http://www.cs.uchicago.edu/groups/theory/erdos.html

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Erdos.html

http://www.oakland.edu/~grossman/erdoshp.html

Paul Erdos, a legendary mathematician who was so devoted to his subject that he lived as a mathematical pilgrim with no home and no job, died Friday in Warsaw, Poland. He was 83.

The cause of death was a heart attack, according to an E-mail message sent out this weekend by Dr. Miki Simonovits, a mathematician at the Hungarian Academy of Sciences, who was a close friend.

Erdos (pronounced AIR-dosh) was attending a mathematics meeting in Warsaw when he died, Simonovits reported.

The news, only now reaching the world's mathematicians, has come as a blow. Dr. Ronald L. Graham, the director of the information sciences research center at AT&T Laboratories, said, "I'm getting E-mail messages from around the world, saying, 'Tell me it isn't so.' "

Never, mathematicians say, has there been an individual like Paul Erdos. He was one of the century's greatest mathematicians, who posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics, which is the foundation of computer science. He was also one of the most prolific mathematicians in history, with more than 1,500 papers to his name. And, his friends say, he was also one of the most unusual.

Erdos, "is on the short list for our century," said Dr. Joel H. Spencer, a mathematician at New York University's Courant Institute of Mathematical Sciences.

Graham said, "He's among the top 10."

Dr. Ernst Straus, who worked with both Albert Einstein and Erdos, wrote a tribute to Erdos shortly before his own death in 1983. He said of Erdos: "In our century, in which mathematics is so strongly dominated by 'theory doctors,' he has remained the prince of problem solvers and the absolute monarch of problem posers."

Erdos, Straus continued, is "the Euler of our time," referring to the great 18th-century mathematician, Leonhard Euler, whose name is spoken with awe in mathematical circles.

Stooped and slight, often wearing socks and sandals, Erdos stripped himself of all the quotidian burdens of daily life: finding a place to live, driving a car, paying income taxes, buying groceries, writing checks. "Property is nuisance," he said.

Concentrating fully on mathematics, Erdos traveled from meeting to meeting, carrying a half-empty suitcase and staying with mathematicians wherever he went. His colleagues took care of him, lending him money, feeding him, buying him clothes and even doing his taxes. In return, he showered them with ideas and challenges -- with problems to be solved and brilliant ways of attacking them.

Dr. Laszlo Babai of the University of Chicago, in a tribute written to celebrate Erdos' 80th birthday, said that Erdos' friends "care for him fondly, repaying in small ways for the light he brings into their homes and offices."

Mathematicians like to brag about their connections to Erdos by citing their "Erdos number." A person's Erdos number was 1 if he or she had published a paper with Erdos. It was 2 if he or she had published with someone who had published with Erdos, and so on.

At last count, Erdos had 458 collaborators, Graham said. An additional 4,500 mathematicians had an Erdos number of 2, Graham added. He said so many mathematicians were still at work on problems they had begun with Erdos that another 50 to 100 papers with Erdos' name on them were expected to be published after his death.

Graham, whose Erdos number is 1, handled Erdos' money for him, setting aside an "Erdos room" in his house for the chore. He said Erdos had given away most of the money he earned from lecturing at mathematics conferences, donating it to help students or as prizes for solving problems he had posed. Erdos left behind only $25,000 when he died, Graham said, and he plans to confer with other mathematicians about how to give it away to help mathematics.

Graham said Erdos' "driving force was his desire to understand and to know." He added, "You could think of it as Erdos' magnificent obsession. It determined everything in his life."

"He was always searching for mathematical truths," said Spencer, of New York University, who also has an Erdos number of 1. He added: "Erdos had an ability to inspire. He would take people who already had talent, that already had some success, and just take them to an entirely new level. His world of mathematics became the world we all entered."

Born in Hungary in 1913, Erdos was a cosseted mathematical prodigy. At age 3, Graham said, Erdos discovered negative numbers for himself when he subtracted 250 degrees from 100 degrees and came up with 150 degrees below zero. A few years later, he amused himself by solving problems he had invented, like how long would it take for a train to travel to the sun.

Erdos had two older sisters who died of scarlet fever a few days before he was born, so his mother became very protective of him. His parents, who were mathematics teachers, took him out of public school after just a few years, Graham said, and taught him at home with the help of a German governess. And, Graham said, Erdos' mother coddled him. "Erdos had never buttered his own toast until he was 21 years old," Graham said. He never married and left no immediate survivors.

When Erdos was 20, he made his mark as a mathematician, discovering an elegant proof for a famous theorem in number theory. The theorem, Chebyshev's theorem, says that for each number greater than one, there is always at least one prime number between it and its double. A prime number is one that has no divisors other than itself and 1.

Although his research spanned a variety of areas of mathematics, Erdos kept up his interest in number theory for the rest of his life, posing and solving problems that were often simple to state but notoriously difficult to solve and that, like Chebyshev's theorem, involved relationships between numbers.

"He liked to say that if you can state a problem in mathematics that's unsolved and over 100 years old, it is probably a problem in number theory," Graham said.

Erdos, like many mathematicians, believed that mathematical truths are discovered, not invented. And he had an evocative way of conveying that notion. He spoke of a Great Book in the sky, maintained by God, that contained the most elegant proofs of every mathematical problem. He used to joke about what he might find if he could just have a glimpse of that book.

He would also muse about the perfect death. It would occur just after a lecture, when he had just finished presenting a proof and a cantankerous member of the audience would have raised a hand to ask, "What about the general case?" In response, Erdos used to say, he would reply, "I think I'll leave that to the next generation," and fall over dead.

Erdos did not quite achieve his vision of the perfect death, Graham said, but he came close.

"He died with his boots on, in hand-to-hand combat with one
more problem. It was the way he wanted to go," Graham said.

Copyright 1996 The New York Times Company

PAUL ERDOS, the mathematician, who has died aged 83, established many records in his field, including the number of papers written (about 1,500), and the number of co-authors (close to 500).

Many mathematicians are eccentric, and Erdös was more so than most. He never had a "proper job"; he had no cheque-book or credit card; he never learnt to drive, and never had health insurance. For most of his life, carrying almost no luggage, wearing sandals and an old suit, he travelled from university to university around the world.

He would bring the mathematical news, pose problems, inspire the locals with his brilliant ideas, and depart in a few days, leaving behind his exhausted hosts to work out the details of their joint work. His open mind, his ability to see the unexpected, and his willingness to wrestle with complications without the help of well-established tools made him a welcome guest wherever he went.

Over the years, Erdös proved important theorems in number theory, geometry, analysis, probability theory, approximation theory, set theory and, above all, combinatorics.

Yet the sophisticated large-scale theories dominating today's mathematics were not to his taste. Rather than build theories, he solved problems - and posed them, often adding spice by offering prizes, ranging from $10 to $10,000, to those who met the challenge.

## After receiving his doctorate in mathematics in 1934, he accepted Mordell's offer of a fellowship at Manchester University

When Erdös failed to find the answers to questions which arose in his field through existing techniques, he would improvise. His "theory" consisted of the accumulation of these original, seemingly unrelated ad hoc methods.

Perhaps his greatest contribution to mathematics was that he realised and demonstrated (decades before it came to be accepted) the importance of chance in finding objects of seemingly contradictory properties, such as efficient networks with few connections. These methods are of paramount importance in computer science, though Erdös himself never touched a computer.

Paul Erdös was born in Budapest on March 26 1913, into a Hungarian-Jewish family. Both his parents were mathematics teachers, and his early education came partly from his mother.

His outstanding work on number theory, undertaken when he was an undergraduate at Budapest University, brought him to the attention of Issai Schur in Berlin, and Louis Mordell at Manchester. After receiving his doctorate in mathematics in 1934, he accepted Mordell's offer of a fellowship at Manchester University. He had planned to go to Germany, but, as he put it, "Hitler got there first."

After four fertile years at Manchester, he left for America, where he was to remain for the next decade. A year at the Institute for Advanced Study in Princeton, when he produced a host of monumental results, was followed by stays at Notre Dame, Purdue, Stanford and other universities.

## Upon receiving $50,000 for his share of the Wolf Prize, in Israel, in 1984, he kept $720 and gave away the rest

With the exception of a nine-year period, when he was not allowed to enter America because of anti-communist hysteria, he spent most of his life there.

Many honours were bestowed on Erdös. Although he did not care about them, he was pleased that they enabled him to help people in need. Upon receiving $50,000 for his share of the Wolf Prize, in Israel, in 1984, he kept $720 and gave away the rest; half the money went to a second cousin he hardly knew, who happened to be in need at that time.

He was a member of many illustrious academies, including the Royal Society and the US National Academy of Science, and he received numerous honorary degrees.

To amuse himself and his friends, Erdos invented a peculiar brand of word-play and imagery. He awarded himself letters after his name: he became PGOM (poor great old man) when his mother died, LD (living dead) at 60, and AD (archaeological discovery) at 65.

Taking his cue from the great English mathematician G H Hardy, he considered God malicious: He gives us colds, hides our glasses and papers, sends us bad weather and traffic jams and, most importantly, is delighted if we fail to do something good when we have a chance.

## A "book-proof" is a thing of beauty; to spite us, God allows us to see one only in exceptional circumstances

As a mathematician in search of beauty, Erdös imagined a book kept by God in which all mathematical theorems are written down, together with their ideal proofs. A "book-proof" is a thing of beauty; to spite us, God allows us to see one only in exceptional circumstances. Erdös himself contributed several book-proofs to the mathematical literature.

Paul Erdös will probably be best remembered for showing that elementary methods (relying on ingenuity rather than vast theories) have a place in contemporary mathematics, and for being the driving force behind the rapid growth of combinatorics.

Paul Erdös lived for mathematics, though he was deeply interested in medicine, history and politics. After his mother's death, he drove himself relentlessly; he slept little and only with the aid of sleeping tablets, and took caffeine pills to help his concentration.

He was unmarried.

© Copyright Telegraph Group Limited 1996.

Friday, September 27 1996; Page A25

The Washington Post

One of the most extraordinary minds of our time has "left." "Left" is the word Paul Erdös, a prodigiously gifted and productive mathematician, used for "died." "Died" is the word he used to signify "stopped doing math." Erdös never "died." He continued doing math, notoriously a young person's field, right until the day he died last Friday. He was 83.

It wasn't just his vocabulary that was eccentric. Erdös's whole life was so improbable no novelist could have invented him (though he was chronicled beautifully by Paul Hoffman in the November 1987 Atlantic Monthly).

He had no home, no family, no possessions, no address. He went from math conference to math conference, from university to university, knocking on the doors of mathematicians throughout the world, declaring, "My brain is open" and moving in. His colleagues, grateful for a few days' collaboration with Erdös -- his mathematical breadth was as impressive as his depth -- took him in.

Erdös traveled with two suitcases, each half-full. One had a few clothes, the other mathematical papers. He owned nothing else. Nothing. His friends took care of the affairs of everyday life for him -- checkbook, tax returns, food. He did numbers.

He seemed sentenced to a life of solitariness from birth, on the day of which his two sisters, age 3 and 5, died of scarlet fever, leaving him an only child, doted upon and kept at home by a fretful mother. Hitler disposed of nearly all the rest of his Hungarian Jewish family. And Erdös never married. His Washington Post obituary ends with this abrupt and rather painful line: "He leaves no immediate survivors."

But in reality he did: hundreds of scientific collaborators and 1,500 mathematical papers produced with them. An astonishing legacy in a field where a lifetime product of 50 papers is considered quite extraordinary.

Mathematicians tend to bloom early and die early. The great Indian genius, Srinivasa Ramanujan, died at 32. The great French mathematician, Evariste Galois, died at 21. (In a duel. The night before, it is said, he stayed up all night writing down everything he knew. Premonition?) And those who don't literally die young, die young in Erdös's sense. By 30, they've lost it.

Erdös didn't. He began his work early. At 20 he discovered a proof for a classic theorem of number theory (that between any number and its double must lie a prime -- i.e., indivisible, number). He remained fecund till his death. Indeed, his friend and benefactor, Dr. (of math, of course) Ron Graham, estimates that perhaps 50 new Erdös papers are still to appear, reflecting work he and collaborators were doing at the time of his death.

Erdös was unusual in yet one other respect. The notion of the itinerant, eccentric genius, totally absorbed in his own world of thought, is a cliche that almost always attaches to the adjective "anti-social." From Bobby Fischer to Howard Hughes, obsession and misanthropy seem to go together.

Not so Erdös. He was gentle, open and generous with others. He believed in making mathematics a social activity. Indeed, he was the most prolifically collaborative mathematician in history. Hundreds of colleagues who have published with him or been advised by him can trace some breakthrough or insight to an evening with Erdös, brain open.

That sociability sets him apart from other mathematical geniuses. Andrew Wiles, for example, recently achieved fame for having solved math's Holy Grail, Fermat's Last Theorem -- after having worked on it for seven years in his attic! He then sprang the proof on the world as a surprise.

Erdös didn't just share his genius. He shared his money. It seems comical to say so because he had so little. But, in fact, it is rather touching. He had so little because he gave away everything he earned. He was a soft touch for whatever charitable or hard-luck cause came his way. In India, he once gave away the proceeds from a few lectures he had delivered there to Ramanujan's impoverished widow.

A few years ago, Graham tells me, Erdös heard of a promising young mathematician who wanted to go to Harvard but was short the money needed. Erdös arranged to see him and lent him $1,000. (The sum total of the money Erdös carried around at any one time was about $30.) He told the young man he could pay it back when he was able to. Recently, the young man called Graham to say that he had gone through Harvard and now was teaching at Michigan and could finally pay the money back. What should he do?

Graham consulted Erdös. Erdös said, "Tell him to do with the $1,000 what I did."

No survivors, indeed.

© Copyright 1996 The Washington Post Company

**PAUL ERDÖS**

**Paul Erdös, mathematician, died on September 20 aged 83. He was born on March 26, 1913. **

Paul Erdös was regarded by fellow mathematicians as the most brilliant, if eccentric, mind in his field. Because he had no interest in anything but numbers, his name was not well known outside the mathematical fraternity. He wrote no best-selling books, and showed a stoic disregard for worldly success and personal comfort, living out of a suitcase for much of his adult life. The money he made from prizes he gave away to fellow mathematicians whom he considered to be needier than himself. "Property is a nuisance," was his succinct evaluation.

Mathematics was his life and his only interest from earliest childhood onwards. He became the most prolific mathematician of his generation, writing or co-authoring 1,000 papers and still publishing one a week in his seventies. His research spanned many areas, but it was in number theory that he was considered a genius. He set problems that were often easy to state, but extremely tricky to solve and which involved the relationships between numbers. He liked to say that if one could think of a problem in mathematics that was unsolved and more than 100 years old, it was probably a problem in number theory.

In spite, or perhaps because of, his eccentricities, mathematicians revered him and found him inspiring to work with. He was regarded as the wit of the mathematical world, the one man capable of coming up with a short, clever solution to a problem on which others had laboured through pages of equations. He collaborated with so many mathematicians that the phenomenon of the "Erdös number" evolved. To have an Erdös number 1, a mathematician must have published a paper with Erdös. To have a number of 2, he or she must have published with someone who had published with Erdös, and so on. Four and a half thousand mathematicians have an Erdös number of 2.

Erdös was born into a Hungarian-Jewish family in Budapest, the only surviving child of two mathematics teachers (his two sisters, who died of scarlet fever, were considered even brighter than he was). At the age of three he was amusing guests by multiplying three-digit numbers in his head, and he discovered negative numbers for himself the same year. When his father was captured in a Russian offensive against the Austro-Hungarian armies and sent to Siberia for six years, his mother removed him from school, which she was convinced was full of germs, and decided to teach him herself. Erdös received his doctorate in mathematics from the University of Budapest, then in 1934 came to Manchester on a post-doctoral fellowship.

By the time he finished there in the late 1930s it was obvious that it would be an act of suicide for a Jew to return to Hungary. Instead Erdös left for the United States. Most members of his family who remained in Hungary were killed during the war.

Erdös had made his first significant contribution to number theory when he was 20, and discovered an elegant proof for the theorem which states that for each number greater than 1, there is always at least one prime number between it and its double. The Russian mathematician Chebyshev had proved this in the 19th century, but Erdös's proof was far neater. News of his success was passed around Hungarian mathematicians, accompanied by a rhyme: "Chebyshev said it, and I say it again/There is always a prime between *n* and *2n*."

In 1949 he and Atle Selberg astounded the mathematics world with an elementary proof of the Prime Number Theorem, which had explained the pattern of distribution of prime numbers since 1896. Selberg and Erdös agreed to publish their work in back-to-back papers in the same journal, explaining the work each had done and sharing the credit. But at the last minute Selberg (who, it was said, had overheard himself being slighted by colleagues) raced ahead with his proof and published first. The following year Selberg won the Fields Medal for his work. Erdös was not much concerned with the competitive aspect of mathematics and was philosophical about the episode.

>From 1954 Erdös began to have problems with the American and Soviet authorities. He was invited to a conference in Amsterdam but on the way back into the United States was interrogated by immigration officials over his Soviet sympathies. Asked what he thought of Marx, he gave a typically guileless response: "I'm not competent to judge, but no doubt he was a great man." Denied his re-entry visa, Erdös left and spent much of the 1950s in Israel.

He was allowed back into the United States in the 1960s, and from 1964 his mother, now in her mid-eighties, began travelling with him. Apart from his family and old friends, Erdös had no interest in a relationship which was not founded in shared intellectual curiosity and he was content to remain a bachelor.

Nor did he see the need to restrict himself to one university. He needed no equipment for his work, no library or laboratory. Instead he criss-crossed America and Europe from one university and research centre to the next, inspired by making new contacts. When he arrived in a new town he would present himself on the doorstep of the local most prominent mathematician and announce: "My brain is open."

He would work furiously for a few days and then move on, once he had exhausted the ideas or patience of his host (he was quite capable of falling asleep at the dinner table if the conversation was not mathematics). He would end sessions with: "We'll continue tomorrow if I live." After the death of his mother in 1971, Erdös threw himself into his work with even greater vigour, regularly putting in a 19-hour day. He fuelled his efforts almost entirely by coffee, caffeine tablets and Benzedrine. He looked more frail, gaunt and unkempt than ever, and often wore his pyjama top as a shirt. Somehow his body seemed to thrive on this punishing routine.

Because of his simple lifestyle, Erdös had little need of money. He won the Wolf Prize in 1983, the most lucrative award for mathematicians, but kept only $720 of the $50,000 he had received. Lecturing fees also went to worthy causes. The only time he required funds was when another mathematician solved a problem which Erdös had set but not been able to solve. From 1954 he had spurred his colleagues on by handing out rewards of up to $1,000 for these problems.

He died from a heart attack at a conference in Warsaw, while he was working on another equation.

Washington Post Staff Writer

Tuesday, September 24 1996; Page B07

The Washington Post

Paul Erdos, 83, one of the world's greatest and most eccentric mathematicians, died Sept. 20 at a hospital in Warsaw after a heart attack. He was stricken while attending a conference.

Dr. Erdos, a Jewish native of Budapest, lived a celibate, monkish and nomadic life devoted to mathematics. He had no home, lived out of a single suitcase and since the 1940s had traveled the world teaching, attending conferences and visiting mathematicians -- he simply stayed with friends the world over.

He was known to arrive, unannounced, at a friend's house with the simple announcement that "my brain is open."

While he was visiting, Dr. Erdos (pronounced AIR-dish) devoted a large part of his time to working with the hosts' mathematics problems, sometimes co-authoring technical articles with them.

It has been said that an above-average mathematician might publish about 20 articles and a really great one 50 in a lifetime. Dr. Erdos, who devoted 19 hours a day, every day, to mathematics, was the author of more than 1,500 works. In 1986, he published 50 papers -- in a field in which it is thought that most peak early.

Dr. Erdos, who was a member of Britain's Royal Society and the national academies on three continents, was known for his work in numbers theory, the theory of sets and probability theory. Over the years, he helped develop such fields of mathematics as random graph theory and combinatorics, mathematics dealing with large numbers of objects that must be counted and classified. He was an inventor of the branch of combinatorics called Ramsey theory.

He was the subject of a prize-winning 1987 profile in the Atlantic magazine and a documentary film of his life recorded in Europe and the United States.

He also had a lifelong fascination for deceptively simple and even ancient branches of mathematics, such as "prime," "perfect" and "friendly" numbers. Simplified, a prime is a number evenly divisible by no other whole number but itself and 1. A perfect number is an integer that equals the sum of other integers that can evenly be divided into it, while pairs of friendly numbers equal the sum of the other's divisors.

Dr. Erdos, who crossed boundaries of **mathematical** study with ease, made his name composing short, pithy and brilliantly simple solutions to problems. Once, while in an instructors lounge, he spotted a problem concerning functional analysis, not his area of expertise. Told that two mathematicians were pleased with a 30-page solution they had arrived at, Dr. Erdos spent about 10 minutes before coming up with a two-line solution.

He was a recipient of the immensely prestigious World Prize, at $50,000 the highest-paying award in mathematics. Despite a spotty income, he gave most of it away, explaining that "some French socialist said that private property was theft" but that he thought "private property is a nuisance."

Dr. Erdos defined the word "mathematician" as "a machine for turning coffee into theorems." Told by many colleagues to slow down, take it easy, he always replied, "There'll be plenty of time to rest in the grave."

Standing 5 feet 6 inches tall and weighing a strapping 130 pounds, he had white hair, glasses and an unruly beard. He lived on a diet consisting largely of caffeine, antidepressants and amphetamines. Some of this might be due to his confession that he had never "learned" to boil water and had not managed to butter his first piece of bread until he was 21. Added to this, a skin condition caused him to wear only silk underwear and led him to wash his hands more than 50 times a day.

Dr. Erdos was born to two mathematics teachers who recognized his gifts early. At the age of 3, he could multiply two three-digit numbers in his head. At 4, he "invented" the concept of negative numbers. He explained that when he was 10 years old, his father explained Euclid's proof to him, and "I was hooked." He entered the University of Budapest as a teenager and left four years later with a doctorate in mathematics.

Dr. Erdos did postgraduate study in England and then found himself wandering the globe as something of a stateless person. As a Jew, he could not live in wartime Europe, and later he was not a fan of communism. To round it all out, he was forbidden to visit the United States during the McCarthy era after explaining to the FBI that he knew nothing of Karl Marx and was interested only in mathematics.

His only known hobby was the Japanese board game go.

He leaves no immediate survivors.

© Copyright 1996 The Washington Post Company