[19847386_21_0] In this lecture we're going to look at the proof of Fermat's Last Theorem. The actual result is something that mathematicians really don't have much use for. I personally don't know of any other mathematical results that rest on Fermat's last theorem. But it's the process of trying to find the proof of this result that would be so important and that would open up entirely new areas of mathematics that would have important repercussions for lots of the applications of mathematics.

If you recall Fermat's last theorem, is something that Fermat referred to in the margin of his copy of Diophantus' Arithmetica. And he'd been looking at the Pythagorean triples. So these again are triples of integers, so that the squares of the first plus the square of the second is equal to the square of the third. Examples of Pythagorean triples are 3, 4, 5; 5, 12, 13; 8, 15, 17 and so on. And we know that there are infinitely many of these. What Fermat had considered was trying to do the same thing for third powers or fourth powers or other higher powers. Can you find three integers so that the first integer cubed plus the second integer cubed is equal to the third integer cubed. Or the first integer to the fourth power plus the second integer to the fourth power is equal to the third integer to the fourth power. Or fifth powers or so on. And he wrote in the margin to Diophantus' Arithmetica that this could not be done but he didn't have a proof that could fit into the margin that he figured out how to do it but he couldn't fit it in there. In fact we do know that Fermat had worked out the case for the exponent four. And so he actually did have that result which was in another manuscript that was discovered later after his death. But the other cases he almost certainly did not know how to prove them. It was in the eighteenth century that Leonhard Euler would take a look at Fermat's last theorem for the exponent three, and come up with a proof that you can't find three integers that satisfy this equation with the exponent three. And as I said when I was talking about Leonhard Euler there are some gaps in his proof. But Euler himself knew the mathematics that was needed to fill in those gaps and so it's appropriate to credit the n equal 3 case, the exponent 3 case of Fermat's last theorem to Leonhard Euler.

So in the beginning of the eighteen hundreds the next case that was out there was the case of the exponent five. And there's a nice story behind proving that there is no solution for the exponent five that involves three very different mathematicians. The first of these was Sophie Germain. And we have talked about her before. She is the person who corresponded with Gauss about number theory. Gauss was so pleased to see a woman working in mathematics. She is the person who discovered the young Galois and what he was trying to do and encouraged him and helped connect him with other mathematicians who might be interested in his work. And Sophie Germain was interested in the problem of Fermat's last theorem and she studied this problem of the exponent five. And while she was not able to find a proof that it could not be done, she was able to show that if there is a solution, then one of those three integers must be divisible by five. This idea was then picked up by the very young Gustav Lejuene Dirichlet. This is in 1825 when Dirichlet began working on this problem he was only twenty years old. We have encountered Dirichlet at various other points later in his career. In 1829 he would publish the very important work on Fourier analysis. And then he would go on and succeed Gauss as chair of mathematics at University of Gottingen in the 1850s. But this was early in his career he was twenty years old, and he was able to show that if there is a solution to Fermat's last theorem, with the exponent [19847386_21_1] five. Then the number that is divisible by five must be an odd integer. Well this then was picked by Adrien-Marie Legendre and we've also encountered Legendre before. He's somebody who played a role in the development of elliptic functions, one of the first people to study this doubly periodic functions. And after Dirichlet did his work Legendre picked it up. This also was in 1825. Legendre at this point was a relatively old man. He was 72. But he was still extremely sharp and he looked at what's Sophie Germain and Gustav Lejeuen Dirichlet had done. They had shown that there must be a number that's divisible by five if there is a solution. Dirichlet had shown that that number must be odd. And what Legendre was able to show is that number can't be odd. The number divisible by five must be even. And since there are no integers that are both even and odd, there can't be a solution with the exponent five.

The next progress that was made on Fermat's last theorem was in 1832. Dirichlet went back to this problem. And he showed that you can't get a solution when the exponent is fourteen. So he's really skipped ahead quite a few and actually you can skip some of the exponents. There's no reason to check the case where the exponent is six. Because if we can't have a solution with the exponent three then we also can't have a solution with the exponent six because if I know that x to the sixth plus y to the sixth is equal to z to the sixth, then I just work with x squared, y squared and z square. And x squared cubed plus y squared cubed is going to equal z squared cubed. And so if I've got a solution with six, I will also have a solution with three. No solution with three means no solution with six. So from here on out we can restrict ourselves to the prime exponents. The next case to study is seven and then after that we need to consider eleven and thirteen and so on through the possible primes. The case of the exponents seven was solved by Gabriel Lame in 1839. And in 1847 Lame took his work on the exponent seven and he announced that he now could extend this and prove Fermat's last theorem in general. And I want to give you some idea of the approach that Lame was taking because it would be very important for some of the later developments in mathematics.

If we back up to the Pythagorean Triples, I want to find three integers x, y, and z so that x squared plus y squared equal z squared. And one of the keys to finding all of the possible triples of integers that give you Pythagorean triples, is to take that polynomial on the left hand side, x squared plus y squared, and factor it. Now if I had x squared minus y squared I could factor that x squared minus y squared is equal to x minus y, times x plus y. X squared plus y squared doesn't seem to factor. But if I introduce the complex integers, so these are integers that are made up of a real part and an imaginary part that also is an integer something like two plus three i, is a complex integer. I can now factor x squared plus y squared, it's equal to x plus i times y, multiplied by x minus i times y. And you use this factorization in order to discover all of the Pythagorean triples. So one of the things that happens though as we begin to expand our idea of integer to what would be called the Gaussian integers. Because Gauss did a lot of work with these complex integers, is that factorization changes. Some of the numbers that are prime remain prime. Three cannot be written as a product of two smaller positive integers, in the ordinary integers. And it also can't be written as a product of two smaller positive integers in the Gaussian integers. But two can be factored. Two is equal to 1 + i multiplied by 1 - i. Five also can be factored. It's no longer a prime in the Gaussian integers. It's 1 + 2i times 1 - 2i. And if you look at the proof that Fermat's last theorem is true for the exponent three, the best way of proving that is actually to use the same sort of idea, you want to try to factor that polynomial x cubed plus y cubed. Now x cubed plus y cubed actually can be factored normally as x plus y multiplied by the quantity x squared minus xy plus y squared. But you can't completely factor it into linear factors. But you can factor it completely into linear factors if you extend your idea of what constitutes an integer and introduce the cube root of one. So the cube root of one which is e to the two pi i over three, if you introduce that and look at the integers that are made up from ordinary integers and then powers of this primitive cube root of one, you can do that factorization of x cubed plus y cubed into linear factors. It turns out to be x plus y, times x plus this cube root of one times y, times x plus the cube root of one squared times y. And using that you can then prove the impossibility of finding a solution to x cubed plus y cubed equal z cubed. And the idea that Lame was using in his work of 1847 was to try to do this factorization in general. If I'm looking at x to the n plus y to the n I want to factor that into these linear factors into n products of simple binomials and if we introduce these integers that use the nth root of unity, then it is possible to do that factorization. Now Lame presented his result to the Academy of Science in Paris in 1847. And there was a good deal of discussion at this engendered. Liouville was one of the mathemticians who was present, we have encountered Liouville before, he is the person who discovered Galois' manuscript [19847386_21_2] saw that it got publicized and saw that Galois theory would actually emerge into the general mathematical conscience. And Liouville pointed out that there was a basic flaw in Lame's approach to this problem of trying to prove Fermat's last theorem. If we think of the primes in the ordinary integers there is only one way of taking an integer and writing it as a product of primes. Six is equal to two times three, or it's equal to three times two. But I can't find two completely different primes and multiply them together in order to get six. If I want to take two primes and multiply them together to get six I've got to use two and three. And Lame had assumed that this would continue to be true for these other kind of extended integers that involve these nth roots of one. And what Liouville said was that he wasn't so sure that that was true that it might in fact be possible that at some point you would encounter cases where factorization was not unique. There might be more than one way of choosing the primes that get multiplied together to get one of these extended integers. Another response that was made to Lame's assertion of the proof was from Cauchy. We've met Cauchy in connection with his attempt to set calculus on a firm foundation. Cauchy had also thought about the problem of Fermat's last theorem. And Cauchy announced Lame that everything that Lame said Cauchy had already thought of first. On March 15th of that year, another mathematician by the name of Wantzel, announced that he had a proof that this factorization in fact would be unique and so Lame's approach should be correct. And the following week on March 22nd both Lame and Cauchy submitted sealed envelopes in which they put all of their work. Neither Lame nor Cauchy felt that they had a complete proof in place yet. They had an idea of proof and so they wanted to write down what they thought they knew and this was a common practice at the Academy of Science at that time, and in many academy of science throughout Europe is that if somebody had a brilliant idea but had not yet worked out all the details they would put that idea down on paper, put it on a sealed envelope and deposit with the academy so that if later on there was a controversy over who came up with the critical idea first we can go back and say open this dated envelope and you'll see that I really knew that result on March 22nd of 1847.

Liouvillle still had his suspicions that perhaps Wantzel wasn't correct, in saying that factorization had to be unique. And in a series of correspondences with another mathematician Ernst Kummer in Berlin, Liouville shared his skepticism, Kummer wrote back and said in fact that your skepticism is well placed. I know for a fact that the factorization is not always going to be unique and as Kummer would later show if you look at the prime exponent 37 and you take extended integers that involve the thirty-seventh root of one, you are not going to get uniqueness in factorization. And so the proof that Lame and also Cauchy thought they had for Fermat's last theorem fell apart at this point. Kummer continued to work on this problem though of the factorization. One of the things he was able to show is that if the factorization is unique, in other words for all of the primes below 37 you can in fact prove Fermat's last theorem. And so he came out with this and he began to study this whole question of these extended integers, these generalizations of the integers something that eventually would come to be called algebraic integers, and what is the property of them, how does that factorization work, how do they differ from the ordinary integers. And this whole work on algebraic integers would really come to its fruition in 1877, when Richard Dedekind would publish a very influential book, The Theory of Algebraic Integers. Richard Dedekind is also somebody that we have encountered before. He was a graduate student with Riemann. He was Riemann's friend who traveled with him in Italy. And it was Richard Dedekind who got many of Riemann's manuscripts published. And one of Dedekind's most important contributions to mathematics would be this great understanding of the algebraic integers. I mentioned that Bernoulli numbers before. These numbers that came out Jakob Bernoulli's work in probability theory. And one of the things that eventually would be [19847386_21_3] discovered as a connection between the numerators of these rational Bernoulli numbers and Fermat's last theorem would eventually would be shown is that if you take the Bernoulli numbers up to the p minus third Bernoulli number, and if p does not divide any of those numerators, then if you look at the algebraic integers that involve the p-th root of one, factorization is going to be unique. So what this says is you take the numerators of the Bernoulli numbers up to p minus three, and if p does not divide any of those numerators, then Fermat's last theorem is true for the exponent p. And eventually people would extend this result, what if p divides one of those numerators but only one of them and p squared does not divide any of those numerators? Eventually it would be possible to extend the result and show that in that case Fermat's last theorem can be proven. Well what if p squared divides one of the numerators but you can't divide p cubed into the product of these numerators? Eventually it would be shown that in that case Fermat's last theorem is true. And so mathematicians began building up cases where they knew Fermat's last theorem was true that you couldn't find the triple of integers so that x to the p plus y to the p equal to z to the p. And by 1976 that it got to the point where was known that Fermat's last theorem was true for all exponents below 100,000. Well that's quite an accomplishment but of course the claim is that Fermat's last theorem is true for all possible exponents. And there are a lot more exponents that are larger than 100,000 than exponents that are less than 100,000.

So a really new approach was needed. And the first breakthrough came about in 1983. The German mathematician Gerd Faltings, who was born in 1954, was able to show that for each exponent greater than or equal to three, we have at most finitely many solutions. If you recall with the Pythagorean triples we actually get infinitely many solutions. And a number of people hope that maybe this would be the door into an ultimate proof. Maybe you could show that once there is one solution, there have to be infinitely many solutions that can be generated from that. And Faltings now has proven that in each case you have at most finitely many solutions. But as it turns out nobody was able to build directly on Faltings result although it was an important result. Something else that was going on at this time as I said Fermat's last theorem had been proven for all of the exponents up to 100,000. And there are many other types of exponents for which it was known that Fermat's last theorem was true. But it was not known that there were infinitely many exponents, for which Fermat's last theorem is true. And in 1985 two mathematicians working quite independently would show that in fact Fermat's last theorem must be true not just for infinitely many exponents, but in some precise mathematical sense it must be true for most of the exponents. There could still be infinitely many exceptions but they would have to occur at rather infrequent intervals. This result was found by Andrew Granville, and Roger Heath-Brown. But even that result would not really lead into the ultimate proof of Fermat's last theorem. What was needed was a totally different approach, and this would come out of the study of elliptic functions and modular functions. And it's the reason why I talk so much about elliptic functions and modular functions in earlier lectures.

I want to go back to Weierstrass' work. I said that Weierstrass this high school teacher had published a very important work on Abelian functions. And one of the things that he accomplished in that paper and in his later work on these Abelian or elliptic functions, was to find a very efficient way of describing these elliptic functions with two specific parameters. Parameters that today we refer to as g2 and g3. You pick two numbers to take on these values g2 and g3 and that uniquely determines your elliptic function. And he denoted these elliptic functions by the German letter p, a script p, and so they are known as p functions today very often instead of elliptic functions, especially if they are in Weierstrass' form. And Weierstrass found the differential equation that these p function satisfy. He was able to show that if you take the derivative of the p function and square it, that is equal to a cubic polynomial in the p function. It's equal to four times the cube of p function, minus this constant [19847386_21_4] g2 times the p function minus the constant g3. Now as succeeding mathematicians studied these elliptic functions, they found that it was very useful to actually look at a surface in which they could be analyzed and the surface that you want to look at is a surface that corresponds to this exact equation, namely it's the surface y squared equals four x cubed minus g2 times x minus g3. Now I said surface, normally that would be a curve because what I have was an algebraic equation between two variables, an x variable and a y variable, and so that defines a curve on the plane. But in this case the input is going to be a complex number and the output is going to be a complex number. And so what you're really looking at is you look at the space of all solutions to this equation, is a two dimensional surface that sits in four dimensional space. This two dimensional surface in four dimensional space is what is known as the elliptic curve. It's called a curve because if we restrict it to one input variable and one output variable, a simple real variable in and a real variable out we would in fact get a curve. But it really is a surface sitting in four dimensional space. And more than that we want to include the point out into infinity and so we really want to look at this elliptic curve and take its projection, in much the same way that I looked at projective space before when I took all of lines through the origin and took each of these lines and replace them by a single point. The same kind of idea can be used with this elliptic curve in order to include the point at infinity in this surface that you're looking at. So you get a very strange surface that you can't actually visualize. It's a two-dimensional surface sitting in four dimensional space to which you would apply this projective operator so that you get the point infinity include it in this particular space. It's called the elliptic curve which is a bit deceptive because it really doesn't have much to do with the ellipses if you recall I was able to trace that adjective elliptic, that's applied to elliptic functions back to something that was connected with the arc length of an ellipse. But the name has stayed there and elliptic curve doesn't look it all like an ellipse. But these elliptic curves turned out to be very useful in many questions, in mathematics, and in analysis, and people began to study these. And you have the elliptic curves that come out of the elliptic functions but there also are other elliptic curves out there. And during the 1950s a number of mathematicians looked at elliptic curves that come from modular functions. So these modular functions of the generalization of theta functions. The modular functions are the functions that are invariant under the kinds of transformations that receive represented in Escher's print the Circle Limit III. And each modular function gives rise to an elliptic curve. And what the mathematicians of the 1950s began to suspect was that every elliptic curve also corresponds to one of these modular functions. If that's true that would be a very powerful statement because it connects two very different areas of mathematics. The geometry of elliptic curves and the analysis of these modular functions. The statement that was made, the conjecture that was made was that every elliptic curve is modular. And this is a conjecture that would be made by two Japanese mathematicians, who were working together, Yutaka Taniyama, and Goro Shimura, and also by a French mathematician who was working in this same area, Andre Weil. And they independently came up with this conjecture that every elliptic curve is modular. And that would be very powerful if we think back back to analytic geometry that provides this correspondence between geometry and algebra. Every algebraic equation gives rise to a geometric curve. Not every geometric curve comes from an algebraic expression. But what if we did know that every algebraic curve or every curve that had sufficiently nice geometric properties in fact corresponded to an algebraic equation? That kind of correspondence would be very useful to know. And what Taniyama Shimura and Weil conjectured is that that precisely is what is going on with the elliptic curves, that every elliptic curve corresponds to a modular function. So if you've got a question about modular functions you can translate that over into the geometry. And it may be obvious what's going on in the geometry and you can use that for insight into the modular functions. Or if you've got a question about the geometry you can translate that into the modular functions and it maybe there's a property of modular functions that enables you to say something important about the geometry.

Now the next progress was made around 1980, when Jean-Pierre Serre and Gerhard Frey were able to show that if Fermat's last theorem is false, that has to give us three particular integers, some x to the n, plus y to the n, equal z to the n. That can be used to construct a very strange elliptic curve. A curve that is so strange that in fact it is not modular. It does not correspond to a modular function. And well they conjectured that this would be the case that was not until 1986, that Ken Ribet was actually able to show that this curve that can be constructed, this geometric object that can be constructed out of a counterexample to Fermat's last theorem actually could not be something that corresponds to a modular function. So if Fermat's last theorem is false, that says that not every elliptic curve is modular. On the other hand if we know that every elliptic curve in fact is modular, that tells us that Fermat's last theorem must be true. And in 1993, Andrew Wiles at Princeton announced that he was able to prove this Taniyama Shimura Weil Conjecture. He actually was not able to prove it in its full generality, but he was able to prove it in the case of the elliptic curves that include this Serre Frey curve that was built on a counterexample to Fermat's last theorem. And so with Andrew Wiles who announced in 1993 that in fact Fermat's last theorem is true, because every elliptic curve of the appropriate kind is modular. There were some flaws in the original proof in 1994, Richard Taylor, one of Andrew Wiles' students was brought in to help of cleaning up the proof and then in 1995 this proof of Fermat's last theorem was finally published.

Initially Fermat's last theorem was done as a special case of the Taniyama Shimura Weil Conjecture, the full Taniyama Shimura Weil Conjecture would be proven in 1999, by a group of mathematicians including Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. Thus closing one of the most important stories in the history of mathematics, a result that was not important in and of itself, but important because of all mathematics to which it gave rise. [End]

Pierre Wantzel: 英文维基。

数学教育数学史数学英语听写

回应转发赞收藏

iced_soda_zyx (Cupertino, United States)

av19847386 听写_Episode 22

Episode 22 Fermat's Last Theorem - The Final Triumph

热门话题 · · · · · · ( 去话题广场 )