An individual's Erdös-Bacon number is the sum of their Erdos number and Bacon number. To have a finite Erdos-Bacon number, therefore, it is necessary (but not always sufficient) to have been involved in a film and contributed to an academic paper. Due to a general lack of mathematicians who act and actors who dabble in hardcore mathematics, the links are allowed to be a bit more tenuous: playing yourself in a film, or appearing merely as an extra, are considered valid ways to achieve your Bacon number, whilst any academic paper (not just in mathematics) can be used to connect with Erdös.
Slashdot favourite Natalie Portman is probably the most famous holder of a finite Erdös-Bacon number: her Erdös number is five via a collaboration chain that starts with a paper from her undergraduate days as a psychology student at Harvard; and she has a Bacon number of one, having appeared alongside Bacon in New York, I Love You, for a total of six.
Erdös himself gets a headstart with his Erdos number of zero; the appearance of Sir Alec Guinness in N is a Number was argued to confer a Bacon number of three, although this role was uncredited and thus the link is overlooked when IMDB-based searches such as the Oracle of Bacon are used. Fortunately, Sam Rockwell's appearance in the more recent Frost/Nixon now provides a chain of length three using only credited roles.
Our own krimson currently has a score of ten, as detailed here.