Paul Erdos 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 "Erdos number" evolved. To have an Erdos number 1, a mathematician must have published a paper with Erdos. To have a number of 2, he or she must have published with someone who had published with Erdos, and so on. Four and a half thousand mathematicians have an Erdos number of 2.
Erdos 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. Erdos 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 Erdos left for the United States. Most members of his family who remained in Hungary were killed during the war.
Erdos 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 Erdos'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 Erdos 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. Erdos was not much concerned with the competitive aspect of mathematics and was philosophical about the episode.
From 1954 Erdos 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, Erdos 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, Erdos 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, Erdos 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, Erdos 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 Erdos 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.