Mathematicians explains Fermat's Last Theorem | Edward Frenkel and Lex Fridman

Would it be useful to perhaps try to explain Oz's Last Theorem? It sure is easy to do. I am optimistic. I guess I always think that everything can be explained. You know, although I say that not everything can be explained. But in mathematics. Be within this particular framework. I think I always feel optimistic when people ask me to explain something. I always start with the assumption that they will understand. Yeah, you know, so let's try Ethereum, one of the mathematics gems of all time. The story also behind this is Pierre Verma, a great French mathematician who lived in the early part of the 17th century and actually has a number of important contributions to his credit, but the most famous one is called the

last

ethereum or return of the firm. a big ethereum and the reason he became so famous is partly because he actually claimed to have proven it himself and he did it in the margin of a book he was reading, which was a really important diafantos book on equations with coefficients and integers and wrote in the margin literally I this equation you know this problem that I will explain in a moment um I have solved it I have found a proof but this margin is too small to contain it at any point.

I give it. I was given a public talk about this and I did it as a joke. I made a tweet in which I wrote that I had proven this

theorem

, but 280 characters is not enough and the type of sentences the client supports is, so this was a 17th century Twitter style proof. Okay, but a lot of

mathematicians

took him seriously because he had great credibility. He made some important contributions and the search continued for 350 years, about 350 years, it remained unproven and many people tried and failed until 1994. No, in 1993. Andrew Wiles announced that a mathematician from Princeton University announced the proof and it was very exciting because he was one of the leading number theorists in the world and unfortunately about a year later a gap was found, so that's exactly what we were talking about before. 99 of the test, this little thing doesn't quite connect and this negates everything, although you could say there are some interesting ideas, but it's not the same as having a test, so apparently I was very frustrated and I was really very worried. people thought it was going to be another 100 years or whatever and then fortunately he was able to enlist with the help of his former student and also the great number theorist Richard Taylor, they were able to do that one percent, so to speak, something Well. people might say maybe not one but five percent or whatever, but it was definitely an important ingredient, but it wasn't.

He had some kind of great new set of ideas and this thing didn't work out, they were able to shut it down. with Taylor and it was finally published and I think it was accepted and refereed in 95 and since it is now believed to be correct, what he actually proved was not Verma's

theorem

itself, but a certain statement which is called the shimurtaniam conjecture, named after three

mathematicians

. Japanese mathematicians and a French porn mathematician who works and also at the Institute for Advanced Study in Princeton and it was my colleague at UC Berkeley Ken Ribbett who in the '80s connected the two problems, that's how it often works in mathematics. to prove statement a instead you prove that a is equivalent to B, so after that, if you can prove B, this would automatically imply that a is correct, so this is what happened here. a was the

last

theorem of form B was a simultaneous conjecture and that's what Andrew Wiles and Richard Taylor actually proved, so to get to Signature's last theorem you require that bridge which was established by my colleague Ken Ribbit at UC Burke.

Now, what is the statement of Hermoslav's theorem? Let me start with Pythagoras, since we already talked about it. I start with the Pythagorean theorem which describes right triangles, so what is the right triangle? It is a triangle in which one of the angles measures 90 degrees, so it has three sides, the longest side is called the hypotenuse and then there are two other sides, so if we denote the high points, the lengths of the hypotenuse by Z and the other two sides X and Y, then Z squared is equal to x squared plus y squared, so that's the equation or main are actually infinite solutions in natural numbers, for example, if x is, you take x is equal to 3 and is equal to 4 and Z is equal to 5. then they solve this equation because 3 3 squared is nine four squared is 16. 9 plus 16 25 and that is 5 squared so x squared plus y squared equals z squared is solved by x equals three y equals four equals five and there are many other solutions of that nature and we should say that natural numbers are integers that are not negative so one, two, three, four, five, six and so on.

What is the last sperm theorem? Firma asked what happens if we replace squares with cubes, for example, so that X Cube plus y Cube equals z Cube. Is there any solution in what you call natural? numbers, it turns out there are none, what about the fourth powers? None or there seems to be none, so that was the statement, so the theorem says that the equation x cubed plus y Cubed equals e cubed has no solutions in natural numbers, eh, remember. natural means positive integers, so of course there is a trivial solution zero zero zeros for this to work, but you need them all to be positive x to the fourth plus y to the fourth is equal to z to the fourth and it also has no solutions uh fifth plus and fifth is equal to z to fifth there are no solutions, so we see the tendency x to N plus and to N plus is equal to z to N if n is greater than two it does not have solutions in natural numbers, that is, one last year's statement from Therma deceptively simple when it comes to famous theorems, you don't need to know anything beyond standard arithmetic addition and multiplication of natural numbers, that's why many people, both specialists and amateurs, try to prove it because it is so easy, so forceful, so easy to formulate, in fact, I think Verma proved the case for cubes.

I think he actually proved something elsewhere in the case of cubes, but it remained a force, there are infinitely many cases, right, even if you prove it for cubes and for fourth power and fifth power, then there are still six sevens and so on. successively in the infinite cases in which it has to be proven and so you see, the simple separate result took 350 years to prove and you know it, but in a sense, it's like mathematicians, you know, you would think that mathematics is such a sterile profession. , everyone's so serious, you know, almost like we're all wearing lab coats and taking an elevator to every tower, and yet, look at all this drama.

In all these dramas, as we all select drama, we also have narratives, we also have our myths. Here is a guy, he is the 16th century mathematician or the 17th century mathematician who lives a note in the margin and motivates others to find the proof, so how many cards were there? broken that they believed they had found the proof and then realized that the proof was incorrect and so on and so on and it takes us to the present and a last attempt and vilification of someone who is a very serious, respected and esteemed mathematician, announces the proof only to be faced with the same reality of their dashed hopes seemingly dashed and like there's a mistake, it doesn't work and then being able to pick up a year later how much drama in this story is incredible, but from what you understand from what you know.

What was the process for him? That's similar to maybe his own life of walking along with the problem for months, not years, yeah, so he worked, he gave interviews about it afterwards, so we know that he described the process of it in that issue. one that he didn't want to tell anyone because he was afraid that people would find out that he was working on it because he was such a high level mathematician that people would guess that he has some idea, that there is some idea, so you know if you know. Someone has an idea, this already gives you a huge confidence boost, so he didn't want people to have that information, so he didn't tell anyone that he was working on it.

Number one. He feels lonely. Number two. He worked on it for seven. years, if I remember correctly, by himself and then he thought he had it and he was elated, he was obviously, you know, very happy and he announced it at the conference, I think it was at the University of Cambridge or the University of Oxford in the United Kingdom in 1993. I think you know, this is really interesting because all of us can really all issues can relate to this because I remember very well my first problem, how I solved my first problem, uh, I described it in love and mathematics in my book, so uh.

How old was he? He was 18 years old. He was a student in Moscow and I was just lucky to be introduced to this great mathematician. You know, I wasn't studying at Moscow University because I encounter Semitism in the Soviet Union, so I was at this technical school, but I was lucky to have a mathematician who took me under his wing and Dmitry Fox, who later came to the US. . USA and is still a professor at UC Davis, not far from me. uh, so he gave me this problem and it was quite technical, so I won't try to describe it, but I do remember how much effort, you know, that excitement but also a kind of fear, what if I don't have what it takes for you?

I know I lost sleep, so this was one of the consequences of this. For the first time in my life I had trouble falling asleep and this guy stayed a couple of years later, so it was like a wake-up call that I should be taking care of myself, not working too late and so on, so it was kind of like that experience and I was lucky to be able to find solution number one in two months. Maybe and it was very surprising and beautiful. I liked it, the answer was in terms of something that seemed to be from A Different World from a different area of ​​mathematics, so I was very happy, but I remember this moment where you suddenly see that, just like you, in this case was literally I had to compile these diagrams with what mathematicians call Co homology groups and spectral sequences and manually calculate some numbers and try to discern some system in them and suddenly I saw that they were all governed by this force, one, one, to a pattern speaks and that was absolutely incredible, so it's like, I mean, what was it?

So you're sitting at a desk, actually you know I lived in a city outside of Moscow, so I used to take a train to Moscow, so it's what we in Russia call electric, you know, like this electric train that was super slow, it took over two hours to cover that distance and I think the crucial revelation came when I was in it and I just had to stop myself. I don't start screaming, you know, because there were other passengers in the car, so I was sitting there looking at this newspaper, so you know what I remember, that's what came to me.

Now I have something that no one else in the world has. a test first of all I didn't do it, it wasn't just the test like in the case of Irma the statement has already been made that's why it's called conjecture you know you make a statement you still don't have proof so you try to prove In my case I didn't know which one would be the answer, there was a type of question where the answer was unknown, so I had to find the answer and try it and the answer was very nice, so no one knew as far as I could tell anyone.

I knew this because my teacher told me that he explored all the literature and this was not known, so I suddenly felt that I was in possession of this now it was a little thing, it was not cured for cancer, you know, it was not a big language. model, you know, but it was something undeniably real and meaningful and it was mine, you know, like no one else had it, I can't publish it, I didn't even tell him, I hadn't even told anyone and it's a very strange feeling , You know? having to have that if you were worried that this treasure might be stolen not at that time not at that time so later there were situations where I was exposed to those types of experiences, but at that time I didn't think about it.

I was still that dreamy kid, you know, who was just obsessed with mathematics, with its Beauty and Discovery, discovering those beautiful facts, beautiful results, so I didn't think, I didn't even think that it might be possible that someone could steal it or whatever. I mean, I just wanted to share it with my teacher as soon as possible, you know, and he understood quickly and said, yeah, good job, you know?