Paul Erdős was a Hungarian mathematician who lived from 1913 to 1996. He was, from a very early age, so utterly engaged in mathematics that he never bothered to learn basic personal and domestic skills, such as how to tie his shoelaces, or how to use a toaster or fry an egg. He lived his life as an academic vagrant, travelling across the world from mathematical conference to conference with all his worldly possessions tucked into a suitcase. He would show up, unannounced, at colleagues' homes, announcing, "My brain is now open!", and would stay just long enough to collaborate on a few papers before moving on.
Erdős drank copious amounts of coffee, and it is this practice of his which may have been his colleague Alfréd Rényi's inspiration for his famous saying, "A mathematician is a machine for turning coffee into theorems". In 1971 Erdős discovered an even stronger stimulant, amphetamine, and took it daily for the last 25 years of his life. His colleagues warned him that the drug was addictive, but he dismissed their claims. One of them bet him $500 that he couldn't stop taking the drug for a month; Erdős won the bet, but complained that his abstinence had set back the progress of mathematics by a month: "Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper." Erdős resumed taking the drug after winning the bet.
Aside from his eccentricities, Erdős is best known as one of the most prolific authors of mathematical papers ever—his total output is second only to the great Leonhard Euler. During his lifetime he authored some 1500 articles with over 500 coauthors. Mathematicians today consider it a badge of pride to have coauthored a paper with Paul Erdős—so much so that they have invented a sort of game out of it. It works like this: If you are Paul Erdős, you have a score, or "Erdős number", of 0. If you coauthored a paper with Erdős, your Erdős number is 1. If you coauthored a paper with someone who coauthored a paper with Erdős, then your Erdős number is 2. And it goes on in this manner indefinitely; in the general case, if you coauthor a paper with someone whose Erdős number is n, then your Erdős number is n + 1. If you cannot link yourself to Paul Erdős by some chain of coauthorship, then you have an Erdős number of infinity.
Recent studies have suggested that most publishing mathematicians have Erdős numbers less than or equal to 15; the median (most common) number is 5, which is slightly greater than the true mean of 4.65. It is considered prestigious to have a very low Erdős number, and not just among mathematicians. Due to the very high frequency of interdisciplinary collaboration in science today, many computer scientists boast a low Erdős number. Erdős number–bearing authors are even found in the social and biological sciences; many linguists, for example, have finite Erdős numbers due to their links with Noam Chomsky, whose number is 4.
Now, before I started my present job in industry, I held various jobs in research and academia, and naturally wrote a few papers. Unlike many scientists, I did most of my research and writing on my own, so I didn't amass a particularly large number of coauthors. Nonetheless, as far back as 1999 I have been able to claim an Erdős number of 6, via my first published paper with Yang Xiang, then of the University of Regina. The chain of coauthorship was as follows:
me → Yang Xiang → Michael Wong → Francis Y. L. Chin → Joseph Y.-T. Leung → Daniel Kleitman → Paul Erdős
Now, in 2007 I coauthored an article with Noam Chomsky, though I didn't know at the time he had an Erdős number of 4. (This I learned only recently when I stumbled across a blog post from Semantics etc.) Through him I could have reduced my own Erdős number from 6 to 5:
me → Noam Chomsky → Marcel-Paul Schützenberger → Samuel Eilenberg → Ivan M. Niven → Paul Erdős
I particularly like this chain because it has lots of famous people (well, famous enough to have Wikipedia articles of their own). However, today, through the help of the AMS Collaboration Distance tool, I have discovered an even shorter path to Erdős which nets me a value equal to Chomsky's 4:
me → Grigoris Antoniou → Cara MacNish → Kaipillil Vijayan → Paul Erdős
Unfortunately, the names on this list aren't nearly so prestigious, but I'm willing to overlook that given that I now have an Erdős number significantly less than average!
If you're interested in learing more about Erdős's strange and wonderful life, I can heartily recommend the very accessible and entertaining biography The Man Who Loved Only Numbers (ISBN 1-85702-829-5) by Paul Hoffman. You don't need to be a mathematician, or even particularly interested in mathematics, to enjoy this book.