The Erdős-Bacon number is defined as the sum of the number of onscreen filmmaking collaborations it takes to connect a person to actor Kevin Bacon, and the number of academic publishing collaborations it takes to reach mathematician Paul Erdős. The number has no value except for the small set of people who are both academics and film performers. Natalie Portman has a Erdős-Bacon number of seven, as does Colin Firth, and Danica McKellar’s number is six. The actor with the lowest number is apparently Albert M. Chan, who appeared with Bacon in Patriots Day. His number is four. Coming from the other direction, Carl Sagan’s number was four. Stephen Hawking’s is seven. The lowest number that anyone is known to have is three, held by Professor Daniel Kleitman of MIT, who was a math advisor for Good Will Hunting (which is one step from Kevin Bacon via Minnie Driver’s appearance in Sleepers) and appeared in the film as an extra.
Can this number be beaten? Lots of mathematicians are still alive who have collaborated with Erdős, and if any of them ever appears in a Kevin Bacon film, they will achieve a value of two. The other way this could be achieved is if Mr. Bacon himself goes into academia and collaborates with one of this group. Since Paul Erdős left us twenty years ago, a value of one is not achievable.
For an even more exclusive club, there are people who have an Erdős-Bacon-Sabbath number, in which the third component is the number of musical collaborations which separate the subject from the members of Black Sabbath. Some famous people for whom low Erdős-Bacon-Sabbath numbers have been claimed include Lisa Kudrow (15), Adam Savage (13), Albert Einstein (11), Richard Feynman (10), Mayim Bialik (10), Tom Lehrer (9), Terry Pratchett (9), Ray Kurzweil (8), and Brian May (8). I don’t think any values lower than eight are known.
Some of these collaborations are a stretch — it’s easy to question whether they count. Among those mentioned earlier, Natalie Portman has a pretty solid 10 via a joke rap track she recorded with some people from Saturday Night Live, Danica McKellar has a dubious 10 by singing in an ad jingle, and Carl Sagan has an even more dubious 10 by being sampled in an autotuned remix of bits of narration from Cosmos. Mayim Bialik’s case might be the poorest of all: Michael Jackson put his celebrity friends into a crowd scene in a music video, and I don’t think she even sings in it. In Brian May’s case, it’s the Erdős side which is very dubious, via some book called Bang! The Complete History of the Universe, which does not exactly seem to be a peer-reviewed publication. For Kurzweil, it’s the Bacon number which is shaky, as they’re counting an appearance on a nonfiction TV show. But some are much more legitimate: for instance, by singing in Mamma Mia!, Colin Firth has given himself a value of 11 which should be beyond dispute.