A conjecture both deep and profound

is whether the circle is round.

In a paper by Erdős,

written in Kurdish,

a counterexample is found.

Today is the 100th birthday of the inordinately prolific, itinerant Hungarian mathematician Paul Erdős (26 March 1913 – 20 September 1996). During his long and illustrious career, he published just over 1500 papers in collaboration with 511 other mathematicians, on a variety of topics (particularly combinatorics, probability, number theory and analysis). It was perhaps inevitable that someone would analyse the interconnectedness of all these collaborations, and thus was born the **Erdős number**: the minimal path length between an author and Erdős in the collaboration graph.

So, Erdős has an Erdős number of 0, someone who has written a paper with Erdős but isn’t themselves Paul Erdős has an Erdős number of 1, someone who’s written a paper with one of them, but who hasn’t written a paper with Erdős himself (and isn’t themselves Paul Erdős) has an Erdős number of 2, and so on. This metric seems to have been first proposed by Caspar Goffman (1913–2006) in an article in a 1969 issue of the *American Mathematical Monthly*.

The American Mathematical Society provides a handy calculator to work out the collaboration distance between any two authors, at least via publications listed in their MathSciNet review database. My own Erdős number is 4, via the following path:

**N M Dunfield**, **S K Friedl**, **N J Jackson**, *Twisted Alexander polynomials of hyperbolic knots*, Experiment. Math. 21 (2012) 329–352

**N M Dunfield**, **D Ramakrishnan**, *Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds*, Amer. J. Math. 132 (2010) 53–97

**V Kumar Murty**, **D Ramakrishnan**, *Period relations and the Tate conjecture for Hilbert modular surfaces*. Invent. Math. 89 (1987) 319–345

**P Erdős**, **M Ram Murty**, **V Kumar Murty**, *On the enumeration of finite groups*, J. Number Theory 25 (1987) 360–378

Further details about the Erdős number can be found here, and biographies of Paul Erdős himself (who was a fascinating person and dedicated mathematician) can be found here, here, or in the book *The Man Who Loved Only Numbers* by Paul Hoffman.

A related concept is that of the **Bacon Number**, the minimal distance between oneself and the actor Kevin Bacon in the appropriate collaboration graph. So, for example, Kevin Bacon’s Bacon number is 0, Jack Nicholson’s is 1 (because they both appeared in *A Few Good Men* (1992)) and so on. The *Oracle of Bacon* provides a useful interface for finding shortest paths between actors listed in the IMDB database.

Depending on how strict you are about these things, my own Bacon number might be 4.

I was in a short amateur film called *G103*, as was a chap called Patrick Niknejad, who at the time was studying for a mathematics degree at Warwick, but who has also acted professionally. In particular, he was in 63 episodes of a children’s television series called *My Parents Are Aliens* which ran from 1999–2006. One of the other regular actors in that series was Tony Gardner, who appeared in *Restoration* (1995) with Meg Ryan, who appeared in *In the Cut* (2003) with Kevin Bacon himself.

Whether or not this actually counts depends on whether or not *G103* is allowed. I argue that it is, because it has been shown on the big screen more than once (at about three of the Warwick Student Cinema‘s regular All-Nighter screenings, as well as a couple of departmental open days). (Also, this is my blog and I’m in charge. So there.)

If you have a finite Bacon number and a finite Erdös number, then their sum provides your Erdös-Bacon number. Relatively few people have a finite Erdös-Bacon number.

The person with the lowest Erdős–Bacon number is probably Steven Strogatz, with an Erdős–Bacon number of 4, according to this. Other notable cases include Carl Sagan (4+2=6), Natalie Portman (5+1=6), Danica McKellar (4+2=6), Brian Greene (3+2=5) and Stephen Hawking (4+2=6).

Something called an Erdős–Bacon–Sabbath number has also been devised – this is the sum of your Erdős number, your Bacon number, and your Sabbath number (how many links between you and Black Sabbath in the musical collaboration graph). The current winner (according to this) is Stephen Hawking, with 4+2+2=8.