Evelyn Lamb: Welcome to My Favorite Theorem, a podcast about math. I’m Evelyn Lamb, one of your cohosts, and I’m a freelance math and science writer in Salt Lake City, Utah.
Kevin Knudson: Hi, I’m Kevin Knudson, a professor of mathematics at the University of Florida. How are you doing, Evelyn? Happy New Year!
EL: Thanks. Our listeners listening sometime in the summer will really appreciate the sentiment. Things are good here. I promised myself I wouldn’t talk about the weather, so instead in the obligatory weird banter section, I will say that I just finished a sewing project, only slightly late, as a holiday gift for my spouse. So that was fun. I made some napkins. Most sewing projects are non-Euclidean geometry because bodies are not Euclidean.
EL: But this one was actually Euclidean geometry, which is a little easier.
KK: Well I’m freezing. No one ever believes this about Florida, but I’ve never been so cold in my life as I have been in Florida, with my 70-year-old, poorly insulated home, when highs are only in the 40s. It’s miserable.
KK: But the beauty of Florida, of course, is that it ends. Next week it will be 75. I’m excited about this show. This is going to be a good one.
EL: Yes, so we should at this point introduce our guest. Today we are very happy to have Ken Ribet on the show. Ken, would you like to tell us a little bit about yourself?
Ken Ribet: Okay, I can tell you about myself professionally first. I’m a professor of mathematics at the University of California Berkeley, and I’ve been on the Berkeley campus since 1978, so we’re coming up on 40 years, although I’ve spent a lot of time in France and elsewhere in Europe and around the country. I am currently president of the American Mathematical Society, which is how a lot of people know me. I’m the husband of a mathematician. My wife is Lisa Goldberg. She does statistics and economics and mathematics, and she’s currently interested in particular in the statistics of sport. We have two daughters who are in their early twenties, and they were home for the holidays.
KK: Good. My son started college this year, and this was his first time home. My wife and I were super excited for him to come home. You don’t realize how much you’re going to miss them when they’re gone.
EL: Hi, Mom! I didn’t go home this year for the holidays. I went home for Thanksgiving, but not for Christmas or New Year.
KK: Well, she missed you.
EL: Sorry, Mom.
KK: So, Ken, you gave us a list of something like five theorems that you were maybe going to call your favorite, which, it’s true, it’s like picking a favorite child. But what did you settle on? What’s your favorite theorem?
KR: Well, maybe I should say first that talking about one’s favorite theorem really is like talking about one’s favorite child, and some years ago I was interviewed for an undergraduate project by a Berkeley student, who asked me to choose my favorite prime number. I said, well, you really can’t do that because we love all our prime numbers, just like we love all our children, but then I ended up reciting a couple of them offhand, and they made their way into the publication that she prepared. One of them is the six-digit prime number 144169, which I encountered early in my research.
KK: That’s a good one.
KR: Another is 1234567891, which was discovered in the 1980s by a senior mathematician who was being shown a factorization program. And he just typed some 10-digit number into the program to see how it would factor it, and it turned out to be prime!
KR: This was kind of completely amazing. So it was a good anecdote, and that reminded me of prime numbers. I think that what I should cite as my favorite theorem today, for the purposes of this encounter, is a theorem about prime numbers. The prime numbers are the ones that can’t be factored, numbers bigger than 1. So for example 6 is not a prime number because it can be factored as 2x3, but 2 and 3 are prime numbers because they can’t be factored any further. And one of the oldest theorems in mathematics is the theorem that there are infinitely many prime numbers. The set of primes keeps going on to infinity, and I told one of my daughters yesterday that I would discuss this as a theorem. She was very surprised that it’s not, so to speak, obvious. And she said, why wouldn’t there be infinitely many prime numbers? And you can imagine an alternative reality in which the largest prime number had, say, 50,000 digits, and beyond that, there was nothing. So it is a statement that we want to prove. One of the interesting things about this theorem is that there are myriad of proofs that you can cite. The best one is due to Euclid from 2500 years ago.
Many people know that proof, and I could talk about it for a bit if you’d like, but there are several others, probably many others, and people say that it’s very good to have lots of proofs of this one theorem because the set of prime numbers is a set that we know a lot about, but not that much about. Primes are in some sense mysterious, and by having some alternative proofs of the fact that there are infinitely many primes, we could perhaps say we are gaining more and more insight into the set of prime numbers.
EL: Yeah, and if I understand correctly, you’ve spent a lot of your working life trying to understand the set of prime numbers better.
KR: Well, so that’s interesting. I call myself a number theorist, and number theory began with very, very simple problems, really enunciated by the ancient Greeks. Diophantus is a name that comes up frequently. And you could say that number theorists are engaged in trying to solve problems from antiquity, many of which remain as open problems.
KR: Like most people in professional life, number theorists have become specialists, and all sorts of quote-on-quote technical tools have been developed to try to probe number theory. If you ask a number theorist on the ground, as CNN likes to say, what she’s working on, it’ll be some problem that sounds very technical, is probably hard to explain to a general listener, and has only a remote connection to the original problems that motivated the study. For me personally, one of the wonderful events that occurred in my professional life was the proof of Fermat’s last theorem in the mid-1990s because the proof uses highly technical tools that were developed with the idea that they might someday shed light on classical problems, and lo and behold, some problem that was then around 350 years old was solved using the techniques that had been developed principally in the last part of the 20th century.
KK: And if I remember right — I’m not a number theorist — were you the person who proved that the Taniyama-Weil conjecture implied Fermat’s Last Theorem?
KR: That’s right. The proof consists of several components, and I proved that something implies Fermat’s Last Theorem.
KR: And then Andrew Wiles partially, with the help of Richard Taylor, proved that something. That something is the statement that elliptic curves (whatever they are) have a certain property called modularity, whatever that is.
EL: It’s not fair for you to try to sneak an extra theorem into this podcast. I know Kevin baited you into it, so you’ll get off here, but we need to circle back around. You mentioned Euclid’s proof of the infinitude of primes, and that’s probably the one most people are the most familiar with of these proofs. Do you want to outline that a little bit? Actually not too long ago, I was talking to the next door neighbors’ 11-year-old kid, he was interested in prime numbers, and the mom knows we’re mathematicians, so we were talking about it, and he was asking about what the biggest prime number was, and we talked about how one might figure out whether there was a biggest prime number.
KR: Yeah, well, in fact when people talk about the proof, often they talk about it in a very circular way. They start with the statement “suppose there were only finitely many primes,” and then this and this and this and this, but in fact, Euclid’s proof is perfectly direct and constructive. What Euclid’s proof does is, you could start with no primes at all, but let’s say we start with the prime 2. We add 1 to it, and we see what we get, and we get the number 3, which happens to be prime. So we have another prime. And then what we do is take 2 and multiply it by 3. 2 and 3 are the primes that we’ve listed, and we add 1 to that product. The product is 6, and we get 7. We look at 7 and say, what is the smallest prime number dividing 7? Well, 7 is already prime, so we take it, and there’s a very simple argument that when you do this repeatedly, you get primes that you’ve never seen before. So you start with 2, then you get 3, then you get 7. If you multiply 2x3x7, you get 6x7, which is 42. You add 1, and you get 43, which again happens to be prime. If you multiply 2x3x7x43 and add 1, you get a big number that I don’t recall offhand. You look for the prime factorization of it, and you find the smallest prime, and you get 13. You add 13 to the list. You have 2, 3, 7, 43, 13, and you keep on going. The sequence you get has its own Wikipedia page. It’s the Euclid-Mullin sequence, and it’s kind of remarkable that after you repeat this process around 50 times, you get to a number that is so large that you can’t figure out how to factor it. You can do a primality test and discover that it is not prime, but it’s a number analogous to the numbers that occur in cryptography, where you know the number is not prime, but you are unable to factor it using current technology and hardware. So the sequence is an infinite sequence by construction. But it ends, as far as Wikipedia is concerned, around the 51st term, I think it is, and then the page says that subsequent terms are not known explicitly.
EL: Interesting! It’s kind of surprising that it explodes that quickly and it doesn’t somehow give you all of the small prime numbers quickly.
KR: It doesn’t explode in the sense that it gets bigger and bigger. You have 43, and it drops back to 13, and if you look at the elements of the sequence on the page, which I haven’t done lately, you’ll see that the numbers go up and then down. There’s a conjecture, which was maybe made without too much evidence, that as you go to the sequence, you’ll get all prime numbers.
EL: Okay. I was about to ask that, if we knew if you would eventually get all of them, or end up with some subsequence of them.
KR: Well, the expectation, which as I say is not based on really hard evidence, is that you should be able to get everything.
KK: Sure. But is it clear that this sequence is actually infinite? How do we know we don’t get a bunch of repeats after a while?
KR: Well, because the principle of the proof is that if you have a prime that’s appeared on the list, it will not divide the product plus 1. It divides the product, but it doesn’t divide 1, so it can’t divide the new number. So when you take the product and you factor it, whatever you get will be a quote-on-quote new prime.
KK: So this is a more direct version of what I immediately thought of, the typical contradiction proof, where if you only had a finite number of primes, you take your product, add 1, and ask what divides it? Well, none of those primes divides it. Therefore, contradiction.
KR: Yes, it’s a direct proof. Completely algorithmic, recursive, and you generate an infinite set of primes.
KK: Okay. Now I buy it.
EL: I’m glad we did it the direct way. Because setting it up as a proof by contradiction when it doesn’t really need the contradiction, it’s a good way, when I’ve taught things like this in the past, this is a good way to get the proof, but you can kind of polish it up and make it a little prettier by taking out the contradiction step since it’s not really required.
KK: And for your 11-year-old friend, contradiction isn’t what you want to do, right? You want a direct proof.
KR: Exactly. You want that friend to start computing.
KK: Are there other direct proofs? There must be.
KR: Well, another direct proof is to consider the numbers known as Fermat numbers. I’ll tell you what the Fermat numbers are. You take the powers of 2, so the powers of 2 are 1, 2, 4, 8, 16, 32, and so on. And you consider those as exponents. So you take 2 to those powers of 2. 2^1, 2^2, 2^4, and so on. To these numbers, you add the number 1. So you start with 2^0, which is 1, 2^1 is 2, and you add 1 and get 3. Then the next power of 2 is 2. You add 1 and you get 5. The next power of 2 is 4. 2^4 is 16. You add 1, and you get 17. The next power of 2 is 8. 2^8 is 256, and you add 1 and get 257. So you have this sequence, which is 3, 5, 17, 257, and the first elements of the sequence are prime numbers. 257 is a prime number. And it’s rather a famous gaffe of Fermat that he apparently claimed that all the numbers in the sequence were prime numbers, that you could just generate primes that way. But in fact, if you take the next one, it will not be prime, and I think all subsequent numbers that have been computed have been verified to be non-prime. So you get these Fermat numbers, a whole sequence of them, an infinite sequence of them, and it turns out that a very simple argument shows you that any two different numbers in the sequence have no common factor at all. And so, for example, if you take 257 and, say, the 19th Fermat number, that pair of numbers will have no common factor. So since 257 happens to be prime, you could say 257 doesn’t divide the 19th Fermat number. But the 19th Fermat number is a big number. It’s divisible by some prime. And you can take the sequence of numbers and for each element of the sequence, take the smallest prime divisor, and then you get a sequence of primes, and that’s a infinite sequence of primes. The primes are all different because none of the numbers have a common factor.
KK: That’s nice. I like that proof.
EL: Nice! It’s kind of like killing a mosquito with a sledgehammer. It’s a big sequence of these somewhat complicated numbers, but there’s something very fun about that. Probably not fun to try to kill mosquitoes with a sledgehammer. Don’t try that at home.
KK: You might need it in Florida. We have pretty big ones.
KR: I can tell you yet a third proof of the theorem if you think we have time.
KR: This proof I learned about, and it’s an exercise in a textbook that’s one of my all-time favorite books to read. It’s called A Classical Introduction to [Modern] Number Theory by Kenneth Irleand and Michael Rosen. When I was an undergraduate at Brown, Ireland and Rosen were two of my professors, and Ken Ireland passed away, unfortunately, about 25 years ago, but Mike Rosen is still at Brown University and is still teaching. They have as an exercise in their book a proof due to a mathematician at Kansas State, I think it was, named Eckford Cohen, and he published a paper in the American Mathematical Monthly in 1969. And the proof is very simple. I’ll tell you the gist of it. It’s a proof by contradiction. What you do is you take for the different numbers n, you take the geometric mean of the first n numbers. What that means is you take the numbers 1, 2, 3, you multiply them together, and in the case of 3, you take the cube root of that number. We could even do that for 2, you take 1 and 2 and multiply them together and take the square root, 1.42. And these numbers that you get are smaller than the averages of the numbers. For example, the square root of 2 is less than 1.5, and the cube root of 6, of 1x2x3, is less than 2, which is the average of 1, 2, and 3. But nevertheless these numbers get pretty big, and you can show using high school mathematics that these numbers approach infinity, they get bigger and bigger. You can show, using an argument by contradiction, that if there were only finitely many primes, these numbers would not get bigger and bigger, they would stop and be all less than some number, depending on the primes that you could list out.
EL: Huh, that’s really cool.
KK: I like that.
KR: That’s kind of an amazing proof, and you see that it has absolutely nothing to do with the two proofs I told you about before.
KK: Well that’s what’s so nice about number theory. It’s such a rich field. You can ask these seemingly simple questions and prove them 10 different ways, or not prove them at all.
KR: That’s right. When number theory began, I think it was a real collection of miscellany. People would study equations one by one, and they’d observe facts and record them for later use, and there didn’t seem to be a lot of order to the garden. And the mathematicians who tried to introduce the conceptual techniques in the last part of the 20th century, Carl Ludwig Siegel, André Weil, Jean-Pierre Serre, and so on, these people tried to make everything be viewed from a systematic perspective. But nonetheless if you look down at the fine grain, you’ll see there are lots of special cases and lots of interesting phenomena. And there are lots of facts that you couldn’t predict just by flying at 30,000 feet and trying to make everything be orderly.
EL: So, I think now it’s pairing time. So on the show, we like to ask our mathematicians to pair their theorem with something—food, beverage, music, art, whatever your fancy is. What have you chosen to pair with the infinitude of primes?
KR: Well, this is interesting. Just as I’ve told you three proofs of this theorem, I’d like to discuss a number of possible pairings. Would that be okay?
KK: Sure. Not infinitely many, though.
KR: Not infinitely many.
EL: Yeah, one for each prime.
KR: One thing is that prime numbers are often associated with music in some way, and in fact there is a book by Marcus du Sautoy, which is called The Music of the Primes. So perhaps I could say that the subject could be paired with his book. Another thing I thought of was the question of algorithmic recursive music. You see, we had a recursive description of a sequence coming from Euclid’s method, and yesterday I did a Google search on recursive music, and I got lots of hits. Another thing that occurred to me is the word prime, because I like wine a lot and because I’ve spent a lot of time in France, it reminds me of the phrase vin primeur. So you probably know that in November there is a day when the Beaujolais nouveau is released all around the world, and people drink the wine of the year, a very fresh young wine with lots of flavor, low alcohol, and no tannin, and in France, the general category of new wines is called vin primeur. It sounds like prime wines. In fact, if you walk around in Paris in November or December and you try to buy vin primeur, you’ll see that there are many others, many in addition to the Beaujolais nouveau. We could pair this theorem with maybe a Côtes du Rhône primeur or something like that.
But finally, I wanted to settle on one thing, and a few days ago, maybe a week ago, someone told me that in 2017, actually just about a year ago, a woman named Maggie Roche passed away. She was one of three sisters who performed music in the 70s and 80s, and I’m sure beyond. The music group was called the Roches. And the Roches were a fantastic hit, R-O-C-H-E, and they are viewed as the predecessors for, for example, the Indigo Girls, and a number of groups who now perform. They would stand up, three women with guitars. They had wonderful harmonies, very simple songs, and they would weave their voices in and out. And I knew about their music when it first came out and found myself by accident in a record store in Berkeley the first year I was teaching, which was 1978-79, long ago, and the three Roches were there signing record albums. These were vinyl albums at the time, and they had big record jackets with room for signatures, and I went up to Maggie and started talking to her. I think I spoke to her for 10 or 15 minutes. It was just kind of an electrifying experience. I just felt somehow like I had bonded with someone whom I never expected to see again, and never did see again. I bought one or two of the albums and got their signatures. I no longer have the albums. I think I left them in France. But she made a big impression on me. So if I wanted to pair one piece of music with this discussion, it would be a piece by the Roches. There are lots of them on Youtube. One called the Hammond Song, is especially beautiful, and I will officially declare that I am pairing the infinitude of primes with the Hammond Song by the Roches.
EL: Okay, I’ll have to listen to that. I’m not familiar with them, so it sounds like a good thing to listen to once we hang up here.
KK: We’ll link it in the show notes, too, so everyone can see it.
EL: That sounds like a lot of fun. It’s always a cool experience to feel like you’re connecting with someone like that. I went to a King’s Singers concert one time a few years ago and got a CD signed, and how warm and friendly people can be sometimes even though they’re very busy and very fancy and everything.
KR: I’ve been around a long time, and people don’t appreciate the fact that until the last decade or two, people who performed publicly were quite accessible. You could just go up to people before concerts or after concerts and chat with them, and they really enjoyed chatting with the public. Now there’s so much emphasis on security that it’s very hard to actually be face to face with someone whose work you admire.
KK: Well this has been fun. I learned some new proofs today.
KR: Fun for me too.
EL: Thanks a lot for being on the show.
KR: It’s my great pleasure, and I love talking to you, and I love talking about the mathematics. Happy New Year to everyone.