Interview with Prof. Ronald Graham

Interview Editorial Consultant: Tai-Ping Liu
Interviewers: Tai-Ping Liu(TPL), Yeong-Nan Yeh(YNY)
Interviewee: Ronald Graham(RG)
Date: October 7th, 2015
Venue: Institute of Mathematics, Academia Sinica

Prof. Ronald Lewis Graham (October 31, 1935 – July 6, 2020) was born at Taft, California. He received his PhD in mathematics in 1962 from the University of California, Berkeley, and then he served at Bell Labs and AT&T Labs in New Jersey, as Director of Information Sciences. In 1999, he left AT&T Labs and moved to the University of California, San Diego (UCSD), as a Professor of Computer and Information Science and also became a chief scientist at the California Institute for Telecommunications and Information Technology. From 1993 to 1994, he was president of the Mathematical Association of America. He has broad perspectives on discrete mathematics and other core pure and applied fields. He is a member of National Academy of Sciences, awarded the American Mathematical Society's Leroy P. Steele Prize for Lifetime Achievement in 2003, and delivered Gibbs Lecture in 2001 and 2015.

TPL: You mention Erdős a lot. May I ask: what made him choose such a lifestyle?

TPL: So you were like a family member taking care of him.

RG: Yes, we had a house with a special Erdős room. He needed his own bathroom. He needed a telephone and access to the refrigerator so he could get food in the middle of the night. There’re many nice stories about that.

TPL: This is really out of respect. You liked him and respected his mathematics. That’s it, right?

RG: Yes, and he had a very good heart. He’s a big heart.

YNY: He’s really good.

RG: There’s a nice story. He was staying with somebody. At four o’clock in the morning, there’s a loud crash in the bathroom. In the morning, he comes to breakfast and didn’t say anything. He eventually said, “You know, there was little accident in your bathroom this morning.There was big bottle of iodine which broke and spilled all over the floor. But don’t worry, I found enough towels to soak it all up”. You may know that it is almost impossible to remove iodine stains. So Erdős meant well, but it didn’t always work out One of his most precious things was the mathematical diary he kept every day, of what he was thinking about. When he died, all those diaries are with Vera T. Sós, his neighbor,and close mathematical colleague. She has them and won’t let anybody see them. We said, “Why?” and Vera said “Well, I just don’t want to”. Many people would like to see what Erdős was thinking. Erdős would write mathematical notes on the right-hand side, and on the left-hand side he would often go back and make another note, such as “Oh, I see, this is a special case of something else I had considered previously.” I would love to see what he was thinking when he was working on the elementary proof of prime number theorem.

TPL: So the diary still exists, but nobody can see it.

RG: Yes, she has them. There are about 20 volumes. I made copies of two of them. But they are in Hungarian, so I couldn’t read them. They had a large meeting for his centennial, few years ago, 800 people attended. I don’t think he was ever in Taiwan.

TPL: Talking about that, my colleague Fon-Che Liu just told me this morning when I mentioned to him about today’s interview. He said that you came to Taiwan rather early on, 1971.

RG: Yes, I did. I spent a summer in Hong Kong. It was the year they opened the first tunnel. Bruce Lee died, I remember. I think I came to National Taiwan University (NTU), giving a talk on scheduling. I think I have a newspaper article about it somewhere. But I was just traveling around. Now in fact, let’s see… Fan-Rong King Chung Graham (金芳蓉) was here, 70-74.

YNY: Yeah, I think she is the classmate of Sun-Yung Alice Chang (張聖容) and Wen-Ch'ing (Winnie) Li (李文卿).

RG: Sun-Yung Alice Chang (張聖容), Wen-Ch'ing (Winnie) Li (李文卿), and 吳徵眉. There was an article by S. S. Chern on the four “golden flowers” in that class. The editor of some mathematics journal said they were going to translate it to English but I don’t think it has happened yet. In fact, there were 5 exceptionally talented women in that class, but one of them died young. So there were really 5 golden flowers.

TPL: In Chinese, it is called 傳記文學. You have met many people and you have worked on many things. Could I ask you one question? What are the things that gave you the most happiness or else were the hardest for you to accomplish?

RG: Well, I mean mathematics is special. When I was little, I really liked astronomy. The stars are really interesting, but it turns out astronomers don’t just look at stars. They don’t look through telescopes but rather use computers to analyze data. But it’s still amazing. People asked me why I do lots of juggling. Many jugglers are in mathematics or computer science, not many people in history or philosophy are jugglers. So what is the connection? The connection seems to be that mathematics is sometimes described as science of patterns. You’re looking for patterns. In juggling, it’s the art of controlling these patterns in time and space. There’s a saying, in juggling, the problem of juggling is that the balls go exactly where you throw them, not where you wish they would go or you hope they would go. In computer programing, the problem is that the computer does exactly what you tell it. There’s no instruction that says, “Well, you know what I mean.” It doesn’t know what you mean. You have to tell it. In mathematics there are unbounded challenges. You never solve all the problems. Every time you write a paper, there’s an extension to ask how about the higher dimensions and so forth. In juggling, there’re always harder and harder tricks. It’s interesting so one of the really difficult tricks is juggling seven balls. Nowadays it is pretty common, the level is just going up. There’s a YouTube video of a juggler juggling 9 balls and then continuing to juggle them while throwing the 9 balls behind his back! Unbelievable! It would seem to be impossible but people are just really determined. And once you see something is possible, then you understand. It’s like the first person run a mile in 4 minutes. That was kind of unbreakable and once it was done, more people did it.

TPL: I remember there’s someone called Bannister or something. It’s rather late, like 70s.

RG: Roger Bannister I think was the first one. Now the record is around 3:45 or so. They now believe that someone can run the marathon in under 2 hours, which is something that seemed impossible a few years ago.

YNY: You were also the professional trampolinist?

RG: Yes, the trampoline. That’s a form of juggling and you are the object. So don’t drop! I moved around a lot when I was young. Every year I went to different schools because my parents worked in the shipyards building ships. So I never had a real high school or junior high school. I skipped a few grades and I didn’t even go to the 12th grade. I went to the University of Chicago when I was 15. Carl Sagan was one of my classmates. It was there that I discovered gymnastics and juggling. At Chicago they had a group that met a several times a week, with classes in various circus skills like juggling, unicycling, gymnastics, etc. They put on shows as a recruiting device to go around the high schools, and show what an interesting place the University of Chicago is. So a part of that was trampoline. I still have a trampoline. Now the level in the world has increased dramatically. Trampoline became an Olympic event when it was introduced in Australian Olympics. Once China saw trampoline as an Olympic event, they wanted to be the best in the world. So China got the best coaches, the best facilities and the best athletes and now they are the best in the world. No question about it. They are the best.

TPL: And they start very young.

RG: Yes, but you need good training, and they have a large population to draw from and good training techniques, and some of the skills they are performing are just breathtaking. Performers often bounce 10 meters above the trampoline. It used to that to spot someone on the trampoline, you stand around the trampoline and when someone flies off, you stand underneath and try to catch them. You don’t do that anymore if someone is coming down from 10 meters. Instead you throw in a protective mat and say, “Good luck!”. Also there are frame pads on the trampoline frame so if you land on them, you should be okay. So if you miss, it’s okay. You don’t get killed. In all of these things there’s a basic principle: To learn a complicated thing, you can break it into the small parts and master the small parts and then put the small parts together. For example, here’s an example. I am working on this cube. When you have a long plane ride, what are the things you’re going to do? So this is a Rubik’s cube.

YNY: This is $9 \times 9 \times 9$?

RG: No, this is $7 \times 7\times 7$. They now make all different sizes. The original Rubik’s cube is 40 years old and that was $3 \times 3\times 3$. And a guy in Greece learned a technique to make bigger ones, up to about 7. But the action wasn’t good. In Mainland China, they figured out a much better construction technique. And now they have sizes up to 17. In these cubes have a lot of pixels on each face. For the centennial meeting of the Mathematical Association of America last summer, I put “MAA” and “100” on the faces of a $13 \times 13 \times 13$ cube and gave it to them. It looks pretty complicated. It’s not that complicated actually. Because the standard technique for this is the following: you first make each face solid. And this is an odd-sized cube, so the center stays at the center. That’s step 1. That takes me about half an hour to actually complete all the faces. Then you make each edge uniform. The edges don’t have to agree with the faces. So pretty soon all the faces are solid and all the edges are uniform. So you can imagine you have a $3 \times 3\times 3$ where the center layer is very fat. So it’s now equivalent to a $3 \times 3\times 3$. Now you can use the algorithm for the 3. So that’s the idea of taking something that looks complicated and reducing it to many small but simpler pieces. So here it’s really mathematical. Normally, as the cube gets more complete, anything you do destroys what you have already done and you don’t want to do that. So you need moves that move a small number of pieces. Here there are various equivalence classes. Any edge piece is not the same as any piece in the inside. I couldn’t put this piece here. These eight pieces in the corners are equivalent. I can put this in any one of the positions, but I couldn’t put this piece here. So these two pieces are equivalent. What you do, for example, say I want to put this piece here. And I am starting to fill the white face. So I am going to put this piece right here. So the idea is this. This is the piece I’m going to put right here. So I am going to put this piece up here, then rotate face and undo what I did. It’s a 3-cycle. So what I’ll do is…so that piece…Now it’s up there. Then I rotate this face and I just put it back. Now I put it underneath here. All I did…it just took all those 3 pieces and permute it, just a 3-cycle.

YNY: I remember 40 years ago when I played the Rubik’s cube, I found that I became a little crazy. I could not stop thinking, just could not. I tried to solve by myself when I was very young. Then I became crazy, I could not stop and I could not sleep, just only thinking about these things. And I found I was going too crazy. My sister and brother told me, “Stop it!” But I could not stop. Finally, I would go to some place outside in the mountain area, yell something like that and run, feel very, very tired and go to sleep, then I stopped.

RG: Well, I am thinking all the time. If you are very brave, you see this piece here. So I think this is the second level down 3, 5 and 6. Well, if you look at this face here, 3, 5, 6 second level. Those three could go right there. There’s one move that makes all three of them at the same time. …3, 5 and 6…go back up, rotate the face and …3, 5 and 6…now I put them there. So let me save a little bit. Once you get the faces, and the edges, and there’s something… What happened is that although it is a group, so you can go from anything to anything, you sometimes end up in an awkward situation. Everything is perfect except one edge has pieces that are a little bit out of order. It is called a parity problem and it only can occur on even-sized cubes. There are complicated moves for dealing with this situation. So just like the trampoline, when you’re looking at the cube, you see everything. While I am looking at it, I just see the pieces that I am planning to move. I don’t see all the other pieces. And now what people can do is amazing. They have speed competitions and a good competitive time is about 8 seconds. But here’s another competition. You take a number of Rubik’s cube and scramble them. You get to look at each one, then you put on a blindfold and try to solve all of them. The current record is 50.

YNY: 50 seconds?

RG: No. 50 cubes! Once you put the blindfold on, you just know now I am at 37, now I am 38, now I am at 39. They count how many you got correct. This YouTube guy shows 49 out of 50, but I think he actually can do 50 out of 50. You have to memorize each one. One question for the $3 \times 3\times 3$ cube is just how many moves are actually necessary to solve any particular scrambled cube? What’s the largest number of moves you’ll ever need? The Google servers just enumerated all possible position and found that you never need more than 20 moves to restore any scrambled cube. In fact, usually 17 moves will suffice, but a few take 20. The speed people don’t use the minimal algorithms since it would take too long to compute them mentally.

YNY: So what’s your opinion about finding the value of the Ramsey number $R(5,5)$? 15 years ago people claimed that this problem will be solved in a few years.

RG: Nobody I know said it was going to be solved. Erdős has a nice story. There’s some alien race came and and demanded that we determine the Ramsey number $R(5,5)$ or they would destroy us. Well, maybe if you get all the people in the world working for a few years, maybe they can do it. But they asked for $R(6,6)$ instead then you just attack them because there no way we could do it.

YNY: That’s a very famous story, yeah.

RG: For example, this cube right here has more than $10^{160}$ arrangements. There’s no way that you could actually enumerate everything and find the minimal solution. You know, $10^{160}$ is way out of anything we can think about computing. Well, in this branch of combinatorics is like a lot of fields. If you can’t solve the original problem, you can generalize it and solve a different special case. For the original Ramsey number problem, you have the complete graph on some number vertices. You would like to color the edges and you got a monochromatic subgraph, of size 5 or 6. So rather than have with two colors of red $K_5$ and blue $K_5$, you might have a red $K_5$ or blue $K_3$. This is the off-diagonal case. And people know more about these and hope that if they can understand these when maybe you could find $R(5, 5)$. Ken Appel and Wolfgang Haken solved the four-color problem by computer in the 70s. Haken feels there will never be a short proof that human beings can read. It is just not in the nature of the problem. It’s true because many special things are true. One of them failed, the theorem would be false. There are systems of arithmetic where you can prove there are theorems in this system which you can state in $n$ symbols but whose shortest proof has length which is doubly exponential in $n$. So you will never be able to write down a proof even though it’s a theorem. There’s a kind of meta-conjecture of number theory, which says if you’d like to prove that a number $n$ is prime, it is going to take the number of symbols that grows at least as fast as $\log n$. If that’s true then for a number like $10^{10^{73}+3}$, you never be able to prove it is prime. You may have a probabilistic test, which says, well, if it’s composite then half of the time it’ll say it is composite and half of the time it doesn’t say anything. It never says it is composite. This is evidence it is prime but it is not a proof. It might be that all the theorems in mathematics we are familiar with are special in that they have proofs so short that human beings can actually write them down in, say, one hundred pages. What about the classification of finite simple groups. You could state the problem in one page, but what is the shortest proof? Can you get it done in a hundred pages ? I don’t think so. A thousand pages? Maybe. For that proof, they split the proof into smaller parts and gave the different parts to different people. Conway was hoping that after they announced the proof, he could find one more simple group that they missed. But he said no, they probably have the complete classification. I think some of the authors were waiting to be the last one. If you are the last one, you could say, and “therefore I finally finished it”. There are results like Kepler’s sphere packing conjecture, accepted by the Annals of Mathematics, but with a disclaimer. They said we had numerous referees look at it but it was so complicated and depends on so much computer computation that they couldn’t completely verify the proof. They were 95 percent convinced that it is correct. I think by now a formal proof has been given. When the computer says it is true, do you believe it ? A hundred-page proof ? I believe in the computer more than a person. Even in the Feit-Thompson paper on groups of odd order, there were a number mistakes in the first version of the manuscript. However, Thompson said, don’t worry, we can fix them. In fact, when Appel and Haken published their paper “Four colors suffice”, they had a follow-up paper, “The four color proof suffices”. The point was, they said, sure, we know the first paper was not perfect, but we were hoping the rest of the community could pitch in and help us get a completely clean proof. We were disappointed they didn’t. By the way, the Four-Color theorem has also been given a formal proof.

TPL: You were at Bell Labs for many years, and that has been one of the important chapters of your mathematical life.

RG: Well, there was quite a strong group at Bell Labs at that time. Bell Labs was part of AT&T which was responsible for the telephone network for the U.S. at that time. Bell Labs was really more like a university but where you don’t have to teach. But you also had real world problems. They invented the transistor, and Claude Shannon, the founder of Information Theory was there. In addition, for example, the Unix operating system was created there. And we had a very strong mathematics group. Fan-Rong King Chung Graham, she was there. I knew her advisor Herb Wilf since Bell Labs was not for from Philadelphia where Wilf was a professor. But as the world changes, there is much more acceleration. When I first came, they told me that perhaps what I worked on might not be of use for the next 10 or 20 years, but that would be fine. They would still be here then and could use the results then. Now a time period like 3 years is considered very long-term. Because for what happens in the computer world, that is two generations. I already have an iPhone 6+, and I am already behind.

YNY: They said that iPhone 6s is the last version. 7 will come soon.

RG: The next generation is usually better but not cheaper and is always right around the corner. So why should you get anything now? Usually we get Apple products but we decided to try an Android phone. We bought a Galaxy 6, pretty nice, pretty good, but it’s a little bit different from an iPhone and has some nice features. So these companies have to compete, get better and better. But it’s a little bit like learning. So when I do symbolic computation I usually use Maple although people also use Mathematica, Matlab, and Sage. The trouble is if you go from one to the other, then it’s a learning process. What is the way to declare a list, string, or a vector or where do you put the parentheses, brackets or braces, and when should you terminate a command with a semicolon or not. It is easier to stay with just one thing, and with Apple that’s one thing. But, you know, it’s a big world so you should experience it. My job at Bell Labs allowed me to travel a lot. So I could take off semesters to teach now and then. I could go to Seton Hall University (which is close to Bell Labs) and study Chinese. So you get exposed to a lot of nice problems because nature works. So there’s something out there that comes from nature and probably it is something that works. For example, we had a problem once, involving lasers. You would like the light to go some medium in the laser, picking up energy when it goes through. The idea was, suppose you put a mirror at each end. The beam can go back and forth many times, picking up lots of energy. But you have to let the beam exit the laser. So you put a small hole at one end so the beam can escape. The beam image consists in small circles at each end. You end up looking for a maximum packing these circles. And we found some very good patterns for these circles. They fabricated it and we hoped to see it in action. The trouble was, the light beam was in the infrared spectrum so we couldn’t see anything! Well, anyway, it worked.

TPL: So math has so many aspects of it. What would be the nature of mathematical research you think in the future?

RG: Well, you know, it’s interesting. In a recent article by Fields Medalist Tim Gowers. he remarks that it is possible that by the year about 2090, all serious mathematics will be done by computers. It will make conjectures, it will find the proofs. And our job as mathematicians is to try to understand and apply some of these results. Now, computers are getting better all the time. In my talk, I mentioned a program called “Graffiti”, which is a program written by an artificial intelligence researcher, and it makes conjectures in graph theory. So far, it has made about six thousand conjectures. In fact, Fan found a nontrivial proof of one of these some years ago. The researcher who wrote the program said the hardest thing is to decide whether a conjecture is interesting or not. Right now computers are not very good at finding proofs. But they pretty good at least at making conjectures. Can you factor a number in polynomial time? And then there is quantum computing. If you build a real quantum computer, you can factor integers in polynomial time. Well, is there any reason you cannot? It seems that initially physicists first thought that there is going to be some natural barrier to being able to build a quantum computer. However now the consensus seems to be that it should be possible. It is just difficult to have enough qubits that are entangled and isolated long enough to be able to do something interesting. Probably a thousand or so would be useful, Right now, systems only have a few dozen. but experimentalists are very clever so we will see what happens in the next few years.

YNY: Zhi-Wei Sun from Nanjing University is a computational number theorist. He made many, many conjectures in combinatorial number theory . He always makes fantastic, ridiculous, those kinds of conjectures about those properties of nth prime .

RG: He does a lot of computer experimentation. He visited California actually for a semester a few years ago. There was a meeting in Vancouver in June. He came as did Chi-Chih Yao and a lot of other people. Even Don Knuth came to spent five days there, which is really unique.

YNY: Just like you said, many people use the computer.

RG: Yes, many, many conjectures are based on the apparent randomness of the primes. And the primes are not really random at all. They have many amazing properties because they are not random, they are actually deterministic. But they have properties that make them appear random. So there’re many conjectures which be obvious if they were actually random. For example, I have a thousand dollar conjecture, namely, you look at the middle $\binom{2n}{n}$. In Pascal’s triangle, it is the entry in the middle of each row. Well, it’s always an even number, which is easy to prove. One question is whether this middle $\binom{2n}{n}$ can be relatively prime to 3. Yes, for example, when $n=3$ since 6 choose 3 is 20. Well, could it be relatively prime to 5? Yes, again, for example for n = 7, 14 choose 7 = 3432 = 4 x 11 x 13 x 17 x 19 which relatively prime to 5 and 7”. The $\$$1000 question is this: Are there infinitely many$n$for which$2n$choose$n$is relatively prime to 3 and 5 and 7 simultaneously?” In other words, it should be relatively prime to 105. Well, I think the answer is yes. The reason I think this is following. For a prime$p$, you might wonder how many times the prime$p$divides this middle$\binom{2n}{n}$? Well, here is the answer. Take$n$and write it to the base$p$, and then add$n$to itself. The number of times you have to carry is the power of$p$that divides$\binom{2n}{n}$. That means if$p$doesn’t divide it, there are no carries. But this means that each digit of$n$base$p$is less than$p/2$. So for 3, it means when you write n to the base 3, it just uses 0’s and 1’s. And in base 5, just 0’s, 1‘s and 2’s and in base 7, just 0’s, 1’s, 2’s and 3’s. So can that happen simultaneously? Well, if you imagine writing a large number in base 3 and base 5, the digits should be independent. Why should the digits in base 3 and base 5 be correlated? You can estimate probabilistically how many$n$up to$x$you expect for all three of these things happen. The answer is$x$to the power around .01. The fact that the power is positive tells you there should be infinitely many, but nobody can prove that. Erdős, Rusza and I showed that this must happen for any two primes. What about for four primes,3, 5, 7 and 11? Now you do that same probabilistic calculation. The number up to$x$that are relatively prime to all those four of these primes is$x$to a small negative number. That says there should be finitely many. Well, 3160 that seems to be the last one. At least it’s the last one until 10 to the 10000th power. Nothing happens between 3160 and$10^{10000}$, so probably that’s the last one. Well, if you believe this random-like behavior, then it’s obvious that it should be happening. If you try to prove it, it is like proving things about normal numbers. A number$n$is normal to the base$b$if all base$b$digits occur (asymptotically) the same number of times, and more generally, all$k$-tuples of base$b$digits occur (asymptotically) the same number of times, and this is true for all$k$. Absolutely normal numbers are those that are normal in every base$b$. Well, almost all numbers are absolutely normal. But nobody actually produced an actual example of single one. It just shows how much more there is for us to know! TPL: It’s like almost all numbers are transcendental. But it is hard to prove that a particular one is. YNY: People really think about these kind of random objects, but later I was told that you are now studying quasi-randomness. What is that? RG: Well, what you like to do was find explicit constructions of objects that behave very much like random objects but yet you know if you can construct them, you understand their properties much better. Suppose you form a large graph on n vertices by choosing each pair of vertices to be an edge independently with probability 1/2. Then certain things are almost certainly true. For example, if you look at half vertices, how many edges you expect? Well, you expect that half of the edges that could be there should be there. Or if you look at the adjacency matrix, which is a 0,1 matrix and you put$(i,j)=1$if there is an edge between$i$and$j$, and 0 otherwise. And you look at the eigenvalues, these symmetric matrices have real eigenvalues, so the largest one should be about at the$n/2$. And the rest should be about$o(n)$. Okay, that’s a quasi-random property. Well, there are dozens of these properties that are equivalent. Once your large graph has one of these properties then it must have them all. So it’s behaving very much like a random graph. The regularity lemma of Szemeredi really says that you can take any large graph and break it up into a bounded number of subgraphs so that almost all of the bipartite graphs formed by pairs of the subgraphs are quasi-random. . In fact the Simons Institute in 2017 had a special half year on pseudo-randomness and quasi-randomness. So Fan and I have been there… TPL: Where? RG: It’s in Berkeley. The Simons Institute for the Theory of Computing. TPL: Well, I think MSRI… RG: Well, MSRI is up on the hill. This is a new one on campus, new campus building. It’s pretty nice staying there for a couple of months last year. James Simons was my classmate in Berkeley, and he earned a lot of money. But he’s putting a lot back in mathematics. Has a nice journal online called Quanta. Do you ever see Quanta? It’s published by Simons Foundation. It’s very nice interviews of people and articles on all kinds of subjects in mathematics, physics and biology. TPL: I can see that you have so much fun with mathematics. Amazing. RG: Well, if it isn’t fun, why are you doing it, right? I tell students, whatever you choose for your career, you had better really like it because whatever you choose, you going be doing it for a long time. TPL: We will translate the interview this into Chinese. And we plan to publish the English version later on also, but that has not happened yet. RG: Do you ever read ICCM Notices? I have the first article in it, a couple of years ago. TPL: By the way, talking about your article, you give a very nice lecture yesterday. Can we have the permission from you to translate it into Chinese and put it into Mathmedia? RG: Well, it doesn’t exist except…it’s just in the air. I think they video-taped it, right? TPL: But you allow us to do that? RG: It’s okay with me. Let me look at it first, though. Because a lot of the stuff is just accumulated over the years and depended on what happened over this time. for example Titang Zhang (張益唐), the record for the prime gap is 246, but the last time I gave the talk it was 270. So yesterday it changed into 246. There’s a website that keeps the current record, but the largest currently known prime is$2^{77, 232, 917}-1\$. However, this seems to change every few years.

TPL: There are conjectures, which are intuitively very easy to grasp and emotionally kind of exciting like the twin prime conjecture. You can tell anybody who knows the multiplication table, basically, and there are other conjectures which take time to explain, and so you were talking about mathematical research in the future. Perhaps one of the things that will survive are the conjectures that is easy to explain and obviously exciting. Like twin primes, there’s a Goldbach conjecture. Say, any even number greater than 2 is a sum of two primes.

YNY: Yes, sum of two primes.

RG: Every number greater than two. The recent result just proved is that every number greater than five is a sum of three primes. People are making progress, but I think to get down to the twin primes you need more. And even Yi-tang Zhang, that was a kind of nice story to me. Although I just read in Wikipedia, he is actually a Professor in Santa Barbara. You see this movie they made about his life. It’s a nice movie, one hour and it’s so peaceful. I like it. And his wife is in California I would say. She’s very energetic, likes to go dancing. I don’t think he’s the dancing type.

TPL: We have had a long chat. So interesting. Thank you very much.

• Tai-Ping Liu and Yeong-Nan Yeh are faculty members at the Institute of Mathematics, Academia Sinica.