Today’s episode of My Favorite Theorem features University of Wisconsin mathematician Jordan Ellenberg. You can listen here or at kpknudson.com. We had a few audio issues in this episode, so if you missed anything, there’s a transcript here.
Ellenberg’s favorite theorem is Fermat’s Little Theorem, a not-so-little result in number theory. It states that if p is a prime number (a positive whole number that is divisible by exactly two distinct factors: itself and 1), and a is any whole number, then ap has a remainder of a when divided by p. As shorthand, mathematicians would say ap is congruent to a mod p. (Mod, short for modulo, just means the remainder when divided by a number. So the number 19 is congruent to 4 mod 5 or 1 mod 3 because it’s 4 more than 15, which is a multiple of 5, and 1 more than 18, which is a multiple of 3.)
Ellenberg stated the theorem in the special case that a=2. Then the theorem becomes the statement that when p is prime, 2p is congruent to 2 mod p, or 2p has a remainder of 2 when divided by p. You can verify this easily for yourself for a few small prime numbers p: 23=8, which has a remainder of 2 when divided by 3. 25=32, which has a remainder of 2 when divided by 5, and so on.
As you’ll hear in the episode, Ellenberg relates Fermat's little theorem to the coefficients of Pascal’s triangle. This triangle (poorly named because it was studied by Persian, Indian, and Chinese mathematicians centuries before Pascal got his hands on it) shows up in a variety of different guises in mathematics. To get the entries in Pascal’s triangle, you start with a 1 by itself in row 0. (Labeling this row 0 instead of 1 makes bookkeeping a little more convenient.) Then each subsequent entry is the sum of the numbers diagonally above it.
Remembering that we start with row 0, the numbers in row n sum up to 2n.
Ellenberg describes how you can look at the entries in prime-numbered rows and practically observe the theorem, at least for the a=2 case, falling out in front of you.
As a resident of Wisconsin, Ellenberg felt a duty to pair his theorem with cheese. He went with gjetost, a caramel-colored Norwegian cheese. You’ll have to listen to the episode to find out why this is the perfect match for Fermat’s Little Theorem.
You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at firstname.lastname@example.org. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.
Previously on My Favorite Theorem: