The idea of six degrees of separation is now quite well known and posits the appealing idea that any two humans on earth are connected by a chain of at most six common acquaintances. In the movie world this idea has become known as the “Bacon number”; for example Elvis Presley has a Bacon number of 2 since he appeared in a film with Edward Asner who in turn appeared in a film with Kevin Bacon.
The mathematical equivalent of the Bacon number is the “Erdős number”. Paul Erdős (1913-1996) was the most prolific mathematician of recent times with more than 1,500 papers, including more than 500 co-authors. The Erdős number now describes how close you are to Paul Erdős in terms of mathematical publications. So, for example, Robin Wilson has an Erdős number of 1 because he co-authored a paper with Erdős, whereas John Watkins has an Erdős number of 2 because he co-authored a paper with Robin Wilson (incidentally he co-authored a paper with Peter Cameron who also has an Erdős number of 1). Even the physicist Albert Einstein had an Erdős number of 2, though this is hardly his greatest claim to fame.
Having a low Erdős number is a matter of great pride for mathematicians. The highest known is 7, although there are also mathematicians who had no connection with Erdős and whose Erdős number is defined as “infinity”. Erdős was so open to working with other mathematicians that it will forever be a deep regret for those of us whose Erdős number is greater than 1 that we never collaborated on a paper with him. Even the great Giancarlo Rota shared this same regret and recalled an evening when he mentioned to Erdős a problem he was working on and Paul provided a hint that eventually led to a complete solution. While Erdős was appropriately thanked in the paper’s introduction, Rota always regretted that he did not include Erdős as a co-author.
Erdős was indeed a genuine mathematical prodigy, and at the age of 19 gave a new and gorgeously simple proof for a well-known theorem about numbers: between any number n and its double 2n there is a prime number. This was his very first mathematical paper.
His supposed obsession with mathematics to the exclusion of anything else in life is now legendary. With no real home base, he traveled the world, staying with friends, visiting math departments, and attending mathematical conferences. He always looked the same, in a suit and a white shirt with an open collar. But, inevitably, he could always be found seated on a couch talking to someone about a mathematical problem.
At conferences he almost always gave a version of a one hour talk he called “open problems” in which, without notes, he would discuss in great detail the current open mathematical problems he was interested in. For many of these problems he would offer monetary rewards for solutions, $100 for a fairly routine problem or perhaps $1,000 or more for a problem he considered especially difficult or important. He knew he could never be able to pay for solutions for all of these problems if they were actually solved, but he also knew that most of them would remain unsolved during his lifetime.
There are countless anecdotes that capture the spirit of Paul Erdős. He could be whimsical: at one conference he announced that he was 81 and most likely a square for the last time. Another time he visited a friend at Santa Clara University in California and upon arrival asked his host “what was the temperature in this valley during the Ice Age?” But, the best stories are from his closest mathematical friends with whom he stayed throughout the years. Many of these have a central theme: at some point in the early morning, about 2 or 3 am, Paul would wander into their bedroom, and with no preamble whatsoever, say something like “about the problem we were discussing last night, what if …”.
The famous neurologist and writer Oliver Sacks said of Paul Erdős: “a mathematical genius of the first order, Paul Erdős was totally obsessed with his subject — he thought and wrote mathematics for nineteen hours a day until the day he died.”
Featured image credit: At the math grad house by kimmanleyort. CC by ND 2.0 via Flickr.