This transcript is provided as a courtesy and may contain errors.
Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, a mathematician at the University of Florida. I’m joined by my other cohost.
Evelyn Lamb: Hi. I’m Evelyn Lamb. I’m a freelance writer currently based in Paris.
KK: Currently based in Paris. For how much longer?
EL: Three weeks. We’re down to the final countdown here. And luckily our bank just closed our account without telling us, so that’s been a fun adventure.
KK: Well, who needs money, right?
KK: You’ve got pastries and coffee, right? So in this episode we are pleased to welcome Jordan Ellenberg, professor of mathematics at the University of Wisconsin. Jordan, want to tell everyone about yourself?
Jordan Ellenberg: Hi. Yes, this is Jordan Ellenberg. I’m talking to you from Madison, Wisconsin today, where we are enjoying the somewhat chilly, drizzly weather we call spring.
KK: Nice. I’ve been to Madison. It’s a lovely place. It’ll be spring for real in a little while, right?
JE: It’ll be lovely. It’s going to be warm this afternoon, and I’m going to be down at the Little League field watching my son play, and it’s as nice as can be.
KK: What position does he play?
JE: He’s 11, so they mix it up. They don’t have defined positions.
KK: I have an 11-year-old nephew who’s a lefty, and they want him to pitch all the time. He’s actually pretty good.
JE: It’s same thing as asking a first-year graduate student what their field is. They should move around a little bit.
KK: That’s absolutely true.
JE: 11 is to baseball as the first year of grad school is to math, I think. Roughly.
KK: That’s about right. Well now they start them so young. We’re getting off track. Never mind. So we’re here to talk about math, not baseball, even though there’s a pretty good overlap there. So Jordan, you’re going to surprise us. We don’t actually know what your favorite theorem is. So why don’t you lay it on us. What’s your favorite theorem?
JE: It is hard to pick your favorite theorem. I think it’s like trying to pick your favorite kind of cheese, though I think in Wisconsin you’re almost required to have one. I’m going to go with Fermat’s Little Theorem.
EL: This is a good theorem. Can you tell us what that is?
JE: I’m not even going to talk about the whole theorem. I’m going to talk about one special case, which I find very beautiful, which is that if you take a prime number, p, and raise 2 to that power, and then you divide by p, then the remainder is 2. In compact terms, you would say 2 to the p is congruent to 2 mod p. Shall we do a couple?
JE: For instance, 2^5 is 32. Computing the remainder when you divide by 5 is easy because you can just look at the last digit. 32 is 2 more than 30, which is a multiple of 5. This persists, and you can do it. Should we do one more? Let’s try. 2 to the 7th is 128, and 126 is a multiple of 7, so 128 is 2 mod 7.
KK: Your multiplication tables are excellent.
JE: Thank you.
KK: I guess being a number theorist, this is right up your alley. Is this why you chose it? How far back does this theorem go?
JE: Well, it goes back to Fermat, which is a long time ago. It goes back very early in number theory. It also goes back for me very early in my own life, which is why I have a special feeling for it. One thing I like about it is that there are some theorems in number theory where you’re not going to figure out how to prove this theorem by yourself, or even observe it by yourself. The way to get to the theorem, and this is true for many theorems in number theory, which is a very old, a very deep subject, is you’re going to study and you’re going to marvel at the ingenuity of whoever could have come up with it. Fermat’s Little Theorem is not like that. I think Fermat’s Little Theorem is something that you can, and many people do, and I did, discover at least that it’s true on your own, for instance by messing with Pascal’s Triangle, for example. It’s something you can kind of discover. At least for me, that was a very formative experience, to be like, I learned about Pascal’s triangle, I was probably a teenager or something. I was messing around and sort of observed this pattern and then was able to prove that 2 to the p was congruent to 2 mod p, and I thought this was great. I sort of told a teacher who knew much more than me, and he said, yeah, that’s Fermat’s Little Theorem.
I was like, “little theorem?” No, this was a lot of work! It took me a couple days to work this out. I felt a little bit diminished. But to give some context, it’s called that because of course there’s the famous Fermat’s Last Theorem, poorly named because he didn’t prove it, so it wasn’t really his theorem. Now I think nowadays we call this theorem, which you could argue is substantially more foundational and important, we call it the little theorem by contrast with the last theorem.
EL: Going back to Pascal’s triangle, I’m not really aware of the connection between Fermat’s Little Theorem and Pascal’s triangle. This is an audio medium. It might be a little hard to go through, but can you maybe explain a little bit about how those are connected?
JE: Sure, and I’m going to gesticulate wildly with my hands to make the shape.
JE: You can imagine a triangle man dance sort of thing with my hands as I do this. So there’s all kinds of crazy stuff you can do with Pascal’s triangle, and of course one thing you can do, which is sort of fundamental to what Pascal’s triangle is, is that you can add up the rows. When you add up the rows, you get powers of two.
JE: So for instance, the third row of Pascal’s triangle is 1-3-3-1, and if you add those up, you get 8, which is a power of 2, it’s 2^3. The fifth row of Pascal’s triangle is 1-5-10-10-5-1. I don’t know, actually. Every number theorist can sort of rattle off the first few rows of Pascal’s triangle. Is that true of topologists too, or is that sort of a number theory thing? I don’t even know.
KK: I’m pretty good.
JE: I don’t want to put you on the spot.
EL: No, I mean, I could if I wrote them down, but they aren’t at the tip of my brain that way.
JE: We use those binomial coefficients a lot, so they’re just like right there. Anyway, 1-5-10-10-5-1. If you add those up, you’ll get 32, which is 2^5. OK, great. Actually looking at it in terms of Pascal’s triangle, why is it the case that you get something congruent to 2 mod 5? And you notice that actually most of those summands, 1-5-10-10-5-1, I’m going to say it a few times like a mantra, most of those summands are multiples of 5, right? If you’re like, what is this number mod 5, the 5 doesn’t matter, the 10 doesn’t matter, the 10 doesn’t matter, the 5 doesn’t matter. All that matters is the 1 at the beginning and the 1 at the end. In some sense Fermat’s Little Theorem is an even littler theorem, it’s the theorem that 1+1=2. That’s the 2. You’ve got the 1 on the far left and the 1 on the far right, and when the far left and the far right come together, you either get the 2016 US Presidential election, or you get 2.
KK: And the reason they add up to powers of 2, I guess, is because you’re just counting the number of subsets, right? The number of ways of choosing k things out of n things, and that’s basically the order of the power set, right?
JE: Exactly. It’s one of those things that’s overdetermined. Pascal’s triangle is a place where so many strands of mathematics meet. For the combinatorists in the room, we can sort of say it in terms of subsets of a set. This is equivalent, but I like to think of it as this is the vertices of a cube, except by cube maybe I mean hypercube or some high-dimensional thing. Here’s the way I like to think about how this works for the case p=3, right, 1-3-3-1. I like to think of those 8 things as the 8 vertices of a cube. Is everybody imagining their cube right now? We’re going to do this in audio. OK. Now this cube that you’re imagining, you’re going to grab it by two opposite corners, and kind of hold it up and look at it. And you’ll notice that there’s one corner in one finger, there’s one corner on your opposite finger, and then the other six vertices that remain are sort of in 2 groups of 3. If you sort of move from one finger to the other and go from left to right and look at how many vertices you have, there’s your Pascal’s triangle, right? There’s your 1-3-3-1.
One very lovely way to prove Fermat’s Little Theorem is to imagine spinning that cube. You’ve got it held with the opposite corners in both fingers. What you can see is that you can sort of spin that cube 1/3 of a rotation and that’s going to group your vertices into groups of 3, except for the ones that are fixed. This is my topologist way. It’s sort of a fixed point theorem. You sort of rotate the sphere, and it’s going to have two fixed points.
EL: Right. That’s a neat connection there. I had never seen Pascal’s triangle coming into Fermat’s little theorem here.
JE: And if you held up a five-dimensional cube with your five-dimensional fingers and held opposite corners of it, you would indeed see as you sort of when along from the corner a group of 5, and then a group of 10, and then a group of 10, and then a group of 5, and then the last one, which you’re holding in your opposite finger.
JE: And you could spin, you could spin the same way, a fifth of a rotation around. Of course the real truth, as you guys know, as we talk about, you imagine a five-dimensional cube, I think everyone just imagines a 3-dimensional cube.
KK: Right. We think of some projection, right?
KK: Right. So you figured out a proof on your own in the case of p=2?
JE: My memory is that I don’t think I knew the slick cube-spinning proof. I think I was thinking of the Pascal’s triangle. This thing I said, I didn’t prove, as we were just discussing, I mean, you can look at any individual row and see that all those interior numbers in the triangle are divisible by 5. But that’s something that you can prove if you know that the elements of Pascal’s triangle are the binomial coefficients, the formula is n!/k!(n-k)!. It’s not so hard to prove in that case that if n is prime, then those binomial coefficients are all divisible by p, except for the first and last. So that was probably how I proved it. That would be my guess.
KK: Just by observation, I guess. Cool.
EL: We like to enjoy the great things in life together. So along with theorems, we like to ask our guests to pair something with this theorem that they think complements the theorem particularly well. It could be a wine or beer, favorite flavor of chocolate…
JE: Since you invited somebody in Wisconsin to do this show, you know that I’m going to tell you what cheese goes with this theorem.
EL: Yes, please.
KK: Yes, absolutely. Which one?
JE: The cheese I’ve chosen to pair with this, and I may pronounce it poorly, is a cheese called gjetost.
JE: Which is a Norwegian cheese. I don’t know if you’ve had it. It almost doesn’t look like cheese. If you saw it, you wouldn’t quite know what it was because it’s a rather dark toasty brown. You might think it was a piece of taffy or something like that.
EL: Yeah, yeah. It looks like caramel.
JE: Yes, it’s caramel colored. It’s very sweet. I chose it because a, because like Fermat’s Little Theorem, I just really like it, and I’ve liked it for a long time; b, because it usually comes in the shape of a cube, and so it sort of goes with my imagined proof. You could, if you wanted to, label the vertices of your cheese with the subsets of a 3-element set and use the gjetost to actually illustrate a proof of Fermat’s Little Theorem in the case p=3. And third, of course, the cheese is Norwegian, and so it honors Niels Henrik Abel, who was a great Norwegian mathematician, and Fermat’s Little Theorem is in some sense the very beginning of what we would now call Abelian group theory. Fermat certainly didn’t have those words. It would be hundreds of years before the general apparatus was developed, but it was one of the earliest theorems proved about Abelian groups, and so in that sense I think it goes with a nice, sweet Norwegian cheese.
EL: Wow, you really thought this pairing through. I’m impressed.
JE: For about 45 seconds before we talked.
EL: I’ve actually made this cheese, or at least some approximation of this. I think it’s made with whey, rather than milk.
JE: On purpose? What happened?
EL: Yeah, yeah. I had some whey left over from making paneer, and so I looked up a recipe for this cheese, and I had never tried the real version of it. After I made my version, then, I went to the store and got the real one. My version stood up OK to it. It didn’t taste exactly the same, but it wasn’t too bad.
KK: Experiments in cheesemaking.
JE: In twelve years, I’ve never made my own cheese. I just buy it from the local dairy farmers.
EL: Well it was kind of a pain, honestly. It stuck to everything. Yeah.
JE: Someone who lives in Paris should not be reduced to making their own cheese, by the way. I feel like that’s wrong.
KK: I’m not surprised you came up with such a good pairing, Jordan. You’ve written a novel, right, years ago, and so you’re actually a pretty creative type. You want to plug your famous popular math book? We like to let people plug stuff.
JE: Yes. My book, which came out here a few years ago, it’s called How Not to Be Wrong. It’ll be out in Paris in two weeks in French. I just got to look at the French cover, which is beautiful. In French it’s called, I’m not going to be able to pronounce it well, like “L’art de ne dire n’importe pas”, [L’art de ne pas dire n’importe quoi] which is “The art of not saying whatever nonsense,” or something like this. It’s actually hard work to translate the phrase “How not to be wrong” in French. I was told that any literal translation of it sounds appallingly bad in French.
This book is kind of a big compendium of all kinds of things I had to say with a math angle. Some of it is about pure math, and insights I think regular people can glean from things that pure mathematicians think about, and some are more on the “statistical news you can use” side. It’s a big melange of stuff.
KK: I’ve read it.
JE: I’m a bit surprised people like it and have purchased it. I guess the publishing house knew that because they wouldn’t have published it, but I didn’t know that. I’m surprised people wanted it.
KK: I own it in hardback. I’ll say it. It’s really well done. How many languages is it into now?
JE: They come out pretty slowly. I think we’ve sold 14 or 15. I think the number that are physically out is maybe . I think I made the book hard to translate by having a lot of baseball material and references to US cultural figures and stuff like that. I got a lot of really good questions from the Hungarian translator. That one’s not out, or that one is out, but I don’t have a copy of it. It just came out.
KK: Very cool.
JE: The Brazilian edition is very, very rich in translator’s notes about what the baseball words mean. They really went the extra mile to be like, what the hell is this guy talking about?
KK: Is it out in Klingon yet?
JE: No, I think that will have to be a volunteer translator because I think the commercial market for Klingon popular math books is not there. I’m holding out for Esperanto. If you want my sentimental favorite, that’s what I would really like. I tried to learn Esperanto when I was kid. I took a correspondence course, and I have a lifelong fascination for it. But I don’t think they publish very many books in Esperanto. There was a math journal in Esperanto.
EL: Oh wow.
KK: That’s right, that’s right. I sort of remember that.
JE: That was in Poland. I think Poland is one of the places where Esperanto had the biggest popularity. I think the guy who founded it, Zamenhof, was Polish.
KK: Cool. This has been fun. Thanks, Jordan.
JE: Thank you guys.
EL: Thanks a lot for being here.
KK: Thanks a lot.
KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at firstname.lastname@example.org. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.