Paul Erdös died on September 20, 1996, at the age of eighty-three. The second most prodigious author of mathematical papers of all time, at his death he had written 1,475 academic research articles. The majority of those papers were co-authored with one or more others. No one has ever come close to Erdös in terms of collaboration: a total of 485 mathematicians wrote at least one joint paper with him. In fact, one way that mathematicians like to measure their importance in the field is by calculating their "Erdös number."

Erdös himself has number 0. Anyone who published a paper with Erdös has Erdös number 1. All those mathematicians who published a paper with someone who has published with Erdös (but have not themselves done so) have Erdös number 2. And so forth. The lower a mathematician's Erdös number, the more important he or she is supposed to be. It is often said that if you don't have an Erdös number, you can't count yourself a "good mathematician." It's not true, of course, but a surprisingly high proportion of mathematicians do indeed have an Erdös number, generally no more than 4 or 5. There is a web site dedicated to providing information on Erdös numbers; it includes a list of all people whose Erdös number is 2 or less.

A key factor behind Erdös's many research collaborations was his highly unusual lifestyle. Though Hungarian by birth, he left his home country in 1934 to avoid political persecution, and spent the rest of his life with no fixed home, wandering the world, for the most part relying on other mathematicians to house and support him. He carried around all his worldly possessions in a shabby suitcase and a plastic bag. Typically, he would show up on the doorstep of a fellow mathematician -- often unannounced -- and declare, "My brain is open," his own peculiar way of inviting the colleague to spend the next few days engaged in a feverish orgy of mathematical investigation.

How did he get away with it? In large part it was his personality. I first met him in 1973, two years after I had completed my Ph.D. in mathematics at the University of Bristol in England. In my thesis, I had answered a question raised many years earlier by Erdös and his Hungarian colleagues. It was hardly an earth-shattering result, but it was enough to get me invited to Hungary to visit the Hungarian Academy of Sciences in Budapest for a week or so, and then travel on to Kesthely for the international conference to celebrate Erdös's sixtieth birthday.

It was hard not to like Erdös. There was a childlike innocence to him, coupled with a wry wit that indicated someone not entirely oblivious to the rest of the world. He was a man whose entire life revolved around mathematics. Short and frail looking, almost cadaverous, he generally dressed in a well-worn gray striped suit, with open sandals, and spoke English with a strong, melodic Hungarian accent. He was able to carry on several mathematical conversations at the same time.

How unusual was Erdös? In 1955, just two year's after Stalin's death, Hungary issued him with a passport that gave him complete freedom to travel in and out of the country. It was the only such passport ever issued, and it was never revoked throughout the entire Cold War period, when other Eastern Europeans found it almost impossible to get exit visas. Moreover, the Hungarian authorities turned a blind eye to the much publicized (by Erdös) fact that he had permanent residency status in Israel -- an "enemy" of Hungary. Erdös's Israeli residency was provided to him when the excesses of Senator Joe McCarthy resulted in his being denied reentry into the United States.

Paul Hoffman's biography tells the story well. Those of us who knew Erdös -- fleetingly in my case -- or even those who merely knew something about him, will enjoy reading the many anecdotes about the man, including descriptions from some of those who regularly opened their homes to him of ways in which he could be a completely infuriating house guest.

Hoffman, formerly editor-in-chief of *Discover* magazine and now the publisher of the Encyclopedia Britannica, first met Erdös in 1986, when he followed the legend around for several weeks, preparing an article for *The Atlantic*. The article won a National Magazine Award. Hoffman followed it up by continuing to track Erdös's activities, and by interviewing a number of mathematicians who knew Erdös well, particularly AT&T's Ron Graham, who, together with his wife Fan Chung, like Graham a world-class mathematician, constructed a guest room at their New Jersey home so that Erdös could stay there whenever he liked.

The book does have some weaknesses, mostly I think because Hoffman is not a mathematician. For instance, one thing the biography does not bring out, but which I think is important, is that Erdös was not a "great mathematician" in the sense of say, Andrew Wiles, who solved Fermat's Last Theorem. Erdös's most significant single result was in fact a new proof of a result that was already known -- the Prime Number Theorem. He did not spend years working on a single, major problem, as Wiles did, nor did he appear to know much mathematics. Erdös's greatest strength was as a "problem solver" who had a rare ability to ask questions that had just the right degree of difficulty. By and large, he preferred those areas of mathematics where all that is required is a sharp mind; he avoided those areas where it is necessary to absorb vast amounts of information in order to make any progress. It was very much an innocent, childlike approach to mathematics. I think this point should have been made -- and explained -- because Erdös did have a greatness, and it was a unique greatness. (It also explains why Erdös was able to collaborate with so many others. On occasion, a single conversation resulted in a joint paper.)

Hoffman also strays way off the main theme on too many occasions. There are, for instance, long discussions of Hardy and Ramanujan, of Ulam, of Godel, of the foundations of mathematics, of Fermat's Last Theorem and Wiles' eventual solution, of the Monty Hall Problem (Marilyn and the Goats), and of Cantor's set theory. Mathematically speaking, these have little to do with Erdös, and I suspect Hoffman included them more because they are all the regular fare of popular expositions of mathematics. They are all good topics, in the right place, but here they not only distract from the main storyline, they also give a picture of mathematics that is different from that portrayed by Erdös himself, and as such will mislead the average reader. The advertising copy for the book claims that Hoffman's aim in including this material is to use Erdös to explain the great mathematical discoveries of all time. In fact, Erdös's extreme atypicality as a mathematician makes this impossible, and the book would have been better for the omission of this material.

But these are the quibbles of a professional. Overall, Hoffman has written an engaging and sympathetic account of Erdös, the man and the mathematician, that should appeal to mathematician and layperson alike, though at times for different reasons.

#### Related Web Sites:

Paul Hoffman has created a web site for his book.

At the Erdos Number Project web site, there is a page containing many links to other material on Erdös.

Keith Devlin is Dean of Science at Saint Mary's College of California and a regular columnist ("Devlin's Angle") for **MAA Online**. His most recent book is Life by the Numbers, the companion book to the PBS television series of the same name, published in April by John Wiley and Sons.