Dr. Dobb's Journal December 1998

Paradigms Past: Paul Erdös

Paul Erdös died in 1996. If you don't know who this legendary mathematician was, this little look back at his life will inform you. I first heard of Paul Erdös from one of my professors in grad school. He had collaborated with Erdös on a paper. Too bad I never wrote a paper with that professor, because -- but I'll explain about Erdös collaborations and Erdös numbers in a moment.

One reason Erdös is legendary is that he was prolific. He holds a record for the most papers written (about 1500), and for the number of coauthors (about 500).

A better reason is that he was brilliant. While a student, he came up with an elegant proof that for every n there is a prime between n and 2n (not the first proof, but the cleverness and elegance of Erdös' proof was typical of his work throughout his life).

But Erdös is perhaps most famous for his eccentricities. To stand out among mathematicians for your eccentricities is quite an accomplishment, but when it came to being peculiar, Erdös was a mathematician's mathematician.

He was a sort of mathematical gypsy, with no fixed home. He would fly from university to university around the world, carrying almost no luggage, stay in the home of one mathematician or another, "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," as they put it in his London Daily Telegraph obituary.

Erdös was also known for offering cash prizes for the solutions to problems he posed, ranging from $10 to $10,000. He won many awards, some of which carried cash grants, but gave the money away freely. When he was given the $50,000 Wolf Prize in Israel, he gave away all but $720, half of it to a needy second cousin he hardly knew.

Erdös' fame and his large number of collaborators have led to a curious sort of game among mathematicians, much like the game involving degrees of separation from actor Kevin Bacon. The game just consists of computing one's Erdös number. Any mathematician has an Erdös number: This is the number of steps the mathematician is from Erdös via collaboration links. For example, all Erdös collaborators have an Erdös number of one; all their collaborators have an Erdös number of two (unless they are also Erdös collaborators, in which case their number is one). Low Erdös numbers are marks of prestige among mathematicians, or at least among those who play the game.

So you see why I wish I had collaborated with the Erdös collaborator. It happens that there is an obscure link between the circle of actors who have acted with Kevin Bacon and the circle of mathematicians who have collaborated with Paul Erdös, so the two games are really just local neighborhoods of a larger game.

For more on Erdös numbers, and to get a start on computing your own Erdös number, visit the Erdös Number Project site (http://www.acs.oakland .edu/~grossman/erdöshp.html), where you will learn that all Fields medal winners through 1998 have Erdös numbers less than six and that at least 63 Nobel prize winners have Erdös numbers less than nine. For more about this extraordinary mathematician, you might read The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth by Paul Hoffman (Hyperion, New York, 1998).

-- M.S.