A person's Erdős–Bacon number is the sum of one's Erdős number—which measures the "collaborative distance" in authoring mathematical papers between that person and Hungarian mathematician Paul Erdős—and one's Bacon number—which represents the number of links, through roles in films, by which the individual is separated from American actor Kevin Bacon. The lower the number, the closer a person is to Erdős and Bacon, and this reflects a small world phenomenon in academia and entertainment.
The idea of Erdős–Bacon numbers has been written about by Simon Singh in the British media and Benjamin Rosenbaum, among others, in the blogosphere.
Roles as self, as a cameo appearance, or as an extra
are often included for the Bacon component. The Erdős criterion
technically refers to collaboration on mathematical papers, but it is
often relaxed to include general research articles for the Erdős–Bacon
In general, to have a defined Erdős–Bacon number, it is necessary
(but not sufficient) for one to have both appeared in a film and
co-authored an academic paper.
Notable scientists with defined Erdős–Bacon numbers include popular astronomer Carl Sagan. One of the best-known actors with a number is actress Natalie Portman,
whose authorship of a psychology paper during her Harvard degree in
psychology earned her an Erdős–Bacon number of 6 (see table below).
It is sometimes reported that Erdős himself has an Erdős–Bacon number
of three. His Erdős number is zero by definition. His Bacon number is
erroneously thought to be three since he appears in N Is a Number: A Portrait of Paul Erdős (1993) with a Gene Patterson, and a Gene Patterson was in Box of Moon Light (1996) with Sam Rockwell who was in Frost/Nixon (2008) with Kevin Bacon. However, this is incorrect, as the Gene Patterson in N Is a Number: A Portrait of Paul Erdős is not the same person as the one in Box of Moon Light (1996).
Bruce Reznick, a professor at University of Illinois at
Urbana-Champaign has an Erdős number of 1 and a Bacon number (by virtue
of being an extra in Pretty Maids All in a Row with Roddy McDowall) of 2, and thus has an Erdős–Bacon number of 3.
For a time, the person with the lowest known Erdős–Bacon number was Dave Bayer, mathematical consultant to A Beautiful Mind who was on screen in a minor role in the movie. Rance Howard was also in A Beautiful Mind and in Apollo 13 with Kevin Bacon to give Bayer a Bacon number of 2. Bayer wrote a paper with Persi Diaconis, who has an Erdős number of 1 due to a jointly authored 1977 Stanford University technical report, later published in a 2004 compilation.
As such, Bayer's Erdős–Bacon number is 4. Diaconis himself has an
Erdős–Bacon number of 5, and Bacon number of 4. He was in the
documentary The Math Life with Freeman Dyson, who was in A Glorious Accident with Oliver Sacks. Sacks has a Bacon number of 2.
Robin E. Harte, former professor of mathematics at University College, Cork, Ireland, has an acting role in Tony Bill's Oliver Twist (1997). Elijah Wood, who appeared with Bacon in Beyond All Boundaries
(2009), plays the Artful Dodger in the film, giving Professor Harte a
Bacon number of 2. Professor Harte has co-authored with Mostafa Mbekhta,
who has co-authored with Endre Makai Jr, who has co-authored with Paul
Erdős, so has an Erdős number of 3. Harte's Erdős–Bacon number is 5.
Astronomer Carl Sagan has an Erdős number of no more than 4 via Steven J. Ostro and a Bacon number of 3, for a total of 7. Physicist Richard Feynman has an Erdős number of 3 and a Bacon number of 3, having appeared in the film Anti-Clock alongside Tony Tang. Physicist H. David Politzer has an Erdős number of 4  and a Bacon number of 2. 
Karl Schaffer is a dancer/choreographer who appeared as a "Killer Klown" in the 1988 film Killer Klowns from Outer Space, and is also a mathematician at De Anza College, with a Bacon number of 2 and an Erdős number of 3, for a sum of 5.
Computer science Professor Marco Pedicini at University of Rome has
an Erdős–Bacon number of 5; he has an Erdős number of 2 (through
Professor Vilmos Komornik) and a Bacon Number of 3 (appearing in the
Italian movie directed by Fausto Brizzi, Notte prima degli esami, Oggi with actress Alessandra Costanzo who played in The Order (2003 film) and this gives her Bacon number of two).
Mathematician and Economics Nobel laureate John Nash, better known in popular culture as the subject of the unauthorized biography A Beautiful Mind and the Academy Award-winning film adaptation has an Erdős number of 4 and a Bacon number of 2, resulting in an Bacon-Erdős number of 6.
Electrical engineer Robert J. Marks II appeared in Ben Stein's movie Expelled: No Intelligence Allowed. Stein appeared with Bacon in Planes, Trains and Automobiles giving Marks a Bacon number of two. Marks has published with Donald C. Wunch II  who published with Frank Harary
who has coauthored with Erdős. Marks therefore has an Erdős number of
three and Erdős–Bacon number of five. Intelligent design proponent William A. Dembski also appeared in Expelled. He has published with Marks  and therefore has a Erdős–Bacon number of six.
Former NCAA gymnastics champion Kiralee Hayashi may be the professional actress with the lowest Erdős number (3), having co-written a peer-reviewed mathematics paper on Riemannian manifolds with Fields medalist Shing-Tung Yau, and having a Bacon number of 2, giving her an Erdős–Bacon number of 5.
Danica McKellar, most famous for her role as Winnie Cooper in The Wonder Years, has an Erdős–Bacon number of 6, having coauthored a mathematics paper published while an undergraduate at UCLA. Her paper gives her an Erdős number of 4, and a Bacon number of 2, both of them having worked with Margaret Easley.
US actress Natalie Portman has an Erdős–Bacon number of 6. She collaborated (using her birth name, Natalie Hershlag) with Abigail A. Baird, who has a collaboration path leading to Joseph Gillis, who has an Erdős number of 1. Bacon and Portman both appear in New York, I Love You, giving Portman a Bacon number of 1 and an Erdős number of 5.
Mayim Bialik has an Erdős–Bacon number of at most 7, having worked on a book chapter and having a 5 point Erdős path  connecting to Shing-Tung Yau. Her Bacon number is 2.
British actor Colin Firth
has an Erdős–Bacon number of 7. Firth is formally credited as co-author
of a neuroscience paper, "Political Orientations Are Correlated with
Brain Structure in Young Adults", after he suggested on BBC Radio 4 that such a study could be done. Another author of that paper, Geraint Rees, has an Erdős number of 5, which gives Firth an Erdős number of 6. Firth appeared with Kevin Bacon in Where the Truth Lies, so his Bacon number is 1.
The movie What the Bleep Do We Know!?, which featured both persons published in the sciences and an actress with Bacon number 2 (Academy Award winner Marlee Matlin), gave Erdős–Bacon numbers to David Albert (Erdős 4,  Erdős–Bacon 7), Fred Alan Wolf (Erdős 5, Erdős–Bacon 8), and Natural Law Party Presidential Candidate John Hagelin (Erdős 5 through frequent collaborator Dimitri Nanopoulos, Erdős–Bacon 8), all appearing as themselves.
Hank Aaron, a baseball
player, is sometimes also considered to have an Erdős–Bacon number of
3, as he and Erdős both autographed the same baseball (for which he is
jokingly referred to as having Erdős number of 1), and he also appeared in Summer Catch with Susan Gardner, who was in In The Cut with Bacon.
For people listed in the Internet Movie Database who are connected to Kevin Bacon, the average Bacon number is 2.957. For mathematicians listed in the American Mathematical Society's MR Collaboration Distance search engine who are connected to Erdős, the average Erdős number is 4.65. There currently exists no exhaustive list of people with defined Erdős–Bacon numbers, but a select group is listed below.
- (b) Includes role as self
- (c) Includes technical report posthumously published in a book (otherwise Erdős number 3, Erdős–Bacon number 5)
- (d) Includes role as extra
- (e) Includes documentary credit
- (g) Includes nonacademic paper
- (h) Includes archival footage
- (i) See discussion above.
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August 14, 2007: "As a UCLA undergraduate, McKellar co-wrote a theorem
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number is 6. Some academics who have had film cameos have lower
Erdos-Bacon numbers [...]"
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