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  • 5/30/2025
In a tale of secrecy, obsession, dashed hopes, and brilliant insights, Princeton math sleuth Andrew Wiles goes undercover for eight years to solve history's most famous math problem: Fermat's Last Theorem. His success was front-page news around the world. But then disaster struck. Re-narrated Horizon episode, "Fermat's Last Theorem"

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00:00Tonight on NOVA
00:02The square of hypotomies is equal to the sum of the squares of the other two sides.
00:09He conquered the impossible.
00:11Suddenly, totally unexpectedly, I had this incredible revelation.
00:15I was flabbergasted, excited, disturbed.
00:19How did this man solve an enigma that mystified the greatest minds for centuries?
00:24I believe I solved Fermat's last theorem.
00:28The proof.
00:50Major funding for NOVA is provided by
00:53The Park Foundation
00:55Dedicated to education and quality television.
01:00And by
01:01The Corporation for Public Broadcasting and viewers like you.
01:05Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion.
01:15One goes into the first room and it's dark, deeply dark, one stumbles around, bumping into the furniture.
01:25And gradually you learn where each piece of furniture is.
01:30And finally, after six months or so, you find the light switch, you turn it on and suddenly it's all illuminated.
01:40You can see exactly where you were.
01:51You can see exactly where you were.
02:01At the beginning of September, I was sitting here, at this desk, when suddenly, totally unexpectedly, I had this incredible revelation.
02:12It was the most, the most important moment of my working life.
02:24Nothing I ever do again will.
02:38I'm sorry.
02:40I'm sorry, that's right.
02:56For seven years, Princeton professor Andrew Wiles worked in complete secrecy, struggling
03:03to solve the world's greatest mathematical problem.
03:08This obsession, which began when he was a child, would later bring him both fame and
03:14regret.
03:15So I came to this, I was a ten-year-old, and one day I happened to be looking in my local
03:24public library, and I found a book on math, and it told a bit about the history of this
03:31problem, that someone had resolved this problem 300 years ago, but no one had ever seen the
03:38proof, no one knew if there was a proof, and people ever since had looked for the proof.
03:43And here was a problem that I, a ten-year-old, could understand, that none of the great mathematicians
03:50in the past had been able to resolve, and from that moment, of course, I just, just tried
03:57to solve it myself.
03:58It was such a challenge, such a beautiful problem.
04:03This problem was Fermat's last theorem.
04:10Pierre de Fermat was, by profession, a lawyer.
04:13He was counsellor to the parliament of Toulouse in France.
04:16But, of course, that's not what he's really remembered for.
04:19What he's really remembered for is his mathematics.
04:22Pierre de Fermat was a 17th century French mathematician who made some of the greatest breakthroughs in
04:31the history of numbers.
04:35His inspiration came from studying the Arithmetica, an ancient Greek text.
04:41Fermat owned a copy of this book, which is a book about numbers, with lots of problems,
04:47which, presumably, Fermat tried to solve.
04:50He studied it.
04:51He, er, he wrote notes in the margins.
04:54Fermat's original notes were lost, but they can still be read in a book published by his son.
05:03It was one of these notes that was Fermat's greatest legacy.
05:08And this is the fantastic observation of Master Pierre de Fermat, which caused all the trouble.
05:14Kubem, autem, endurus, kubus.
05:17This tiny note is the world's hardest mathematical problem.
05:21It's been unsolved for centuries.
05:24Yet it begins with an equation so simple that children know it by heart.
05:29The square of the hypotenuse is equal to the sum of the squares of the other two sides.
05:37Yeah, well, that's Pythagoras's theorem, isn't it?
05:39That's what we all did at school.
05:41So, Pythagoras's theorem, the clever thing about it is that it tells us when three numbers are the sides of a rectangle triangle.
05:50That happens just when x squared plus y squared equals z squared.
05:56x squared plus y squared equals z squared.
06:01And you can ask, well, what are the whole number solutions of this equation?
06:05You quickly find there's a solution.
06:083 squared plus 4 squared equals 5 squared.
06:11Another one is 5 squared plus 12 squared is 13 squared.
06:17And you go on looking and you find more and more.
06:20So then the natural question is, the question Fermat raised,
06:25supposing you change from squares, supposing you replace the 2 by 3, by 4, by 5, by 6, by any whole number n.
06:36And Fermat said simply that you'll never find any solutions.
06:43However, however far you look, you'll never find a solution.
06:47If n is greater than 2, you will never find numbers that fit this equation.
06:56That's what Fermat said.
06:58What's more, he said he could prove it.
07:00But instead, he scribbled a most enigmatic note.
07:04Written in Latin, he says he has a truly wonderful proof, demonstrationem mirabilem, of this fact.
07:12And then the last words are, hank marginis exiguatis non caprit.
07:17This margin is too small to contain it.
07:23So Fermat said he had a proof, but he never said what it was.
07:29Fermat made lots of marginal notes.
07:33People took them as challenges.
07:35And over the centuries, every single one of them has been disposed of.
07:39And the last one to be disposed of is this one.
07:42That's why it's called the last theorem.
07:45Rediscovering Fermat's proof became the ultimate challenge.
07:49A challenge which would baffle mathematicians for the next 300 years.
07:54Gauss, the greatest mathematician in the world.
07:57Oh yeah, Galois.
07:59Kummer, of course.
08:01Well, in the 18th century, Euler's didn't prove it.
08:04Well, you know, there's only been the one woman, really.
08:06Sophie Germain.
08:08Oh, there are millions of women.
08:10There are lots of people in there.
08:12But nobody had any idea where to start.
08:14Well, mathematicians just love a challenge.
08:20And this problem, this particular problem just looks so simple.
08:24It just looked as if it had to have a solution.
08:27And of course, it's very special because Fermat said he had a solution.
08:31This thing's been there like a beacon in front of us.
08:34I mean, if you give up, you just get the feeling you've given up.
08:37It's like Everest.
08:38It won't go away.
08:39It still stays there.
08:40And so one person can give up, but another person is still just trying to get a little bit further.
08:49The task was to prove that no numbers other than two fit the equation.
08:55But when computers came along, couldn't they check each number one by one and show that none of them worked?
09:06Well, how many numbers are there to be dealt with?
09:08You've got to do it for infinitely many numbers.
09:11So after you've done it for one, how much closer have you got?
09:15Well, there's still infinitely many left.
09:17After you've done it for a thousand numbers, how many, how much closer have you got?
09:20Well, there's still infinitely many left.
09:22In fact, you haven't done very many, have you?
09:29A computer can never check every number.
09:32Instead, what's needed is a mathematical proof.
09:37A mathematician is not happy until the proof is complete and considered complete by the standards of mathematics.
09:45In mathematics, there's the concept of proving something, of knowing it with absolute certainty.
09:49Which, well, it's called rigorous proof.
09:52Well, the rigorous proof is a series of arguments.
09:56Based on logical deductions.
09:58Which just, um, build one upon another.
10:02Step by step.
10:03Until you get to, um...
10:05A complete proof.
10:06That's what mathematics is about.
10:08A proof provides a logical demonstration of why no numbers fit the equation without having to check every number.
10:20After centuries of failing to come up with such a proof, mathematicians began to abandon Fermat.
10:27In the seventies, Fermat was no longer in fashion.
10:45At the same time, Andrew Wiles was just beginning his career as a mathematician.
10:51He went to Cambridge University as a research student under the supervision of Professor John Coates.
10:58I've been very fortunate to have Andrew as a student.
11:01And even as a research student, he, uh, he was a wonderful person to work with.
11:06He had very deep ideas then, and, uh, it, it was always clear he was a mathematician who would do great things.
11:17But not with Fermat.
11:19Everyone thought Fermat's last theorem was impossible.
11:22So Professor Coates encouraged Andrew to forget his childhood dream and work on more mainstream math.
11:31The problem with working on Fermat is that you could spend years getting nothing.
11:37It's fine to work on any problem so long as it generates mathematics.
11:40The, almost the definition of good mathematical problem is the mathematics it generates rather than the problem itself.
11:48Uh, you know, not all mathematical problems are useless.
11:51Fermat's one really is useless, I think, in a certain sense.
11:54It's got no practical value whatsoever.
11:56If it's true, it doesn't imply anything profound that any of us know.
12:00Uh, it doesn't lead to anything that's useful that any of us know.
12:04Uh, it's, uh, by itself is sort of on the outskirts.
12:10It's not what you would consider a mainstream, important, central question in modern mathematics.
12:17And that point I really put aside Verma.
12:20Uh, it's not that I forgot about it.
12:23It was always there, always remembered it.
12:25But I realized that the only techniques we had to tackle it had been around for 130 years.
12:32And it didn't seem they were really getting to the root of the problem.
12:38So when I went to Cambridge, uh, my advisor, John Coates, was working on Iwasawa theory and elliptic curves, and I started working with him.
12:47For Andrew's advisor and a host of other mathematicians, elliptic curves were the in thing to study.
13:00You may never have heard of elliptic curves, but they're extremely important.
13:08Okay, so what's an elliptic curve?
13:10Elliptic curves.
13:11They're not ellipses.
13:12They're cubic curves whose solution have a shape that looks like a donut.
13:19It looks so simple, yet the complexity, especially arithmetic complexity, is immense.
13:32Every point on the donut is the solution to an equation.
13:36Andrew Wiles now studied these elliptic equations and set aside his dream.
13:42What he didn't realize was that on the other side of the world, elliptic curves and Fermat's last theorem were becoming inextricably linked.
14:04I entered the University of Tokyo in 1949, and that was four years after the war.
14:14But almost all professors were tired, and the lectures were not inspiring.
14:26Goro Shimura and his fellow students had to rely on each other for inspiration.
14:32In particular, he formed a remarkable partnership with a young man by the name of Yutaka Tanayama.
14:40That was when I became very close to Tanayama.
14:46Tanayama was not a very careful person as a mathematician.
14:54He made a lot of mistakes, but he made mistakes in a good direction.
15:04And so eventually he got the right answers.
15:10And I tried to imitate him, but I found out that it is very difficult to make good mistakes.
15:22Together, Tanayama and Shimura worked on the complex mathematics of modular functions.
15:28I really can't explain what a modular function is in one sentence.
15:34I can try and give you a few sentences to explain.
15:38I really can't do it in one sentence.
15:40Oh, it's impossible.
15:42There's a saying attributed to Eichler that there are five fundamental operations of arithmetic.
15:48Addition, subtraction, multiplication, division, and modular forms.
15:54Modular forms are functions on the complex plane that are inordinately symmetric.
16:04They satisfy so many internal symmetries that their mere existence seem like accidents, but they do exist.
16:16This image is merely a shadow of a modular form.
16:19To see one properly, your TV screen would have to be stretched into something called hyperbolic space.
16:25Bizarre modular forms seem to have nothing whatsoever to do with the humdrum world of elliptic curves.
16:37But what Tanayama and Shimura suggested shocked everyone.
16:43In 1955, there was an international symposium.
16:51And Tanayama posed two or three problems.
16:59The problems posed by Tanayama led to the extraordinary claim that every elliptic curve was really a modular form in disguise.
17:09It became known as the Tanayama-Shimura conjecture.
17:17But the Tanayama-Shimura conjecture says, it says that every rational elliptic curve is modular.
17:23And that's so hard to explain.
17:29So let me explain.
17:31Over here, you have the elliptic world.
17:34The elliptic curves, these donuts.
17:36And over here, you have the modular world.
17:39Modular forms with their many, many symmetries.
17:43The Shimura-Tanayama conjecture makes a bridge between these two worlds.
17:50These worlds live on different planets.
17:54It's a bridge. It's more than a bridge. It's really a dictionary.
18:00A dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.
18:13I think that when Shimura and Tanayama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous.
18:21I wasn't studying mathematics yet.
18:24By the time I was a graduate student in 1969 or 1970, people were coming to believe the conjecture.
18:33In fact, Tanayama-Shimura became a foundation for other theories which all came to depend on it.
18:39But Tanayama-Shimura was only a conjecture, an unproven idea.
18:46And until it could be proven, all the mathematics which relied on it were under threat.
18:52We built more and more conjectures, stretched further and further into the future.
18:57But they would all be completely ridiculous if Tanayama-Shimura was not true.
19:10Proving the conjecture became crucial.
19:13But tragically, the man whose idea inspired it didn't live to see the enormous impact of his work.
19:20In 1958, Tanayama committed suicide.
19:23I was very much puzzled.
19:36Puzzled meant maybe the best word.
19:39Of course I was sad, but you see, it was so sudden.
19:45And I was unable to make sense out of this.
19:49Well, some people suggest that he lost confidence in himself.
20:05That may be so, but I think it was more complex.
20:10I don't really know.
20:11Confidence in himself, but not mathematically.
20:23Taniyama-Shimura went on to become one of the great unproven conjectures.
20:35A foundation for many important mathematical ideas.
20:39But what did it have to do with Fermat's last theorem?
20:42At that time, no one had any idea that Tanayama-Shimura could have anything to do with Fermat.
20:51Of course, in the 80s, that all changed completely.
20:54But what was the bridge between the two ideas?
21:07Tanayama-Shimura says every elliptic curve is modular.
21:12And Fermat says no numbers fit this equation.
21:16What was the connection?
21:17Well, on the face of it, the Shimura-Taniyama conjecture, which is about elliptic curves, and Fermat's last theorem have nothing to do with each other.
21:43Because there's no connection between Fermat and elliptic curves.
21:48But in 1985, Gerhard Frey had this amazing idea.
21:53Frey, a German mathematician, considered the unthinkable.
21:57What would happen if Fermat was wrong, and there was a solution to this equation after all?
22:05Frey showed how starting with a fictitious solution to Fermat's last equation, if such a horrible beast existed,
22:11he could make an elliptic curve with some very weird properties.
22:16That elliptic curve seems to be not modular.
22:20But Shimura-Taniyama says that every elliptic curve is modular.
22:24So if there is a solution to this equation, it creates such a weird elliptic curve, it defies Tanayama-Shimura.
22:31So in other words, if Fermat is false, so is Shimura-Taniyama.
22:37Or, said differently, if Shimura-Taniyama is correct, so is Fermat's last theorem.
22:42Fermat and Tanayama-Shimura were now linked, apart from just one thing.
22:48The problem is that Frey didn't really prove that his elliptic curve was not modular.
22:54He gave a plausibility argument, which he hoped could be filled in by experts,
22:58and then the experts started working on it.
23:03In theory, you could prove Fermat by proving Tanayama, but only if Frey was right.
23:10Frey's idea became known as the Epsilon Conjecture, and everyone tried to check it.
23:15One year later, in San Francisco, there was a breakthrough.
23:21I saw Barry Mazur on the campus, and I said, let's go for a cup of coffee,
23:26and we sat down for cappuccinos at this cafe.
23:30And I looked at Barry, and I said, you know, I'm trying to generalize what I've done
23:35so that we can prove the full strength of Sayre's Epsilon Conjecture.
23:39And Barry looked at me and said, but you've done it already.
23:41All you have to do is add on some extra gamma zero of M structure and run through your argument,
23:47and it still works, and that gives everything you need.
23:49And this had never occurred to me, as simple as it sounds.
23:52I looked at Barry, I looked at my cappuccino, I looked back at Barry, and I said, my God, you're absolutely right.
23:57Ken's idea was brilliant.
23:58And I was completely enthralled, I just sort of wandered back to my apartment in a cloud,
24:03and I sat down and I ran through my argument, and it worked, it really worked.
24:06And at the conference, I started telling a few people that I'd done this,
24:10and soon large groups of people knew, and they were running up to me, and they said,
24:13is it true that you've proved the Epsilon Conjecture?
24:15And I had to think for a minute, and all of a sudden, I said, yes, I have.
24:19I was at a friend's house sipping iced tea early in the evening,
24:35and he just mentioned casually in the middle of a conversation,
24:39by the way, do you hear that Ken has proved the Epsilon Conjecture?
24:43And I was just electrified, I knew that moment.
24:46the course of my life was changing, because this meant that,
24:54to prove Feynman's Last Theorem, I just had to prove Tanyama Shimura Conjecture.
25:01From that moment, that was what I was working on.
25:05I just knew I would go home and work on the Tanyama Shimura Conjecture.
25:09Andrew abandoned all his other research.
25:20He cut himself off from the rest of the world, and for the next seven years,
25:24he concentrated solely on his childhood passion.
25:26I never use a computer. I sometimes scribble, I do doodles.
25:42I start trying to find patterns, really.
25:46So I'm doing calculations which try to explain some little piece of mathematics.
25:53I'm trying to fit it in with some previous, broad, conceptual understanding of some branch of mathematics.
26:02Sometimes that'll involve going and looking up in a book to see how it's done there.
26:09Sometimes it's a question of modifying things a bit.
26:13Sometimes doing a little extra calculation.
26:15And sometimes you realize that nothing that's ever been done before is any use at all.
26:20And you just have to find something completely new.
26:22And it's a mystery where it comes from.
26:27I must confess, I did not think that the Shimura-Tanyama conjecture was accessible to proof at present.
26:34I thought I probably wouldn't see a proof in my lifetime.
26:37I was one of the vast majority of people who believed that the Shimura-Tanyama conjecture was just completely inaccessible.
26:43And I didn't bother to prove it, even think about trying to prove it.
26:45Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you can actually go and prove this conjecture.
26:53In this case, certainly, the first several years I had no fear of competition.
26:58I simply didn't think I or anyone else had any real idea how to do it.
27:03But I realized after a while that talking to people casually about Fermat was impossible because it just generates too much interest and you can't really focus yourself for years unless you have this kind of undivided concentration, which too many spectators would have destroyed.
27:29Andrew decided that he would work in secrecy and isolation.
27:37I often wondered myself what he was working on.
27:40Didn't have an inkling.
27:41No, I suspected nothing.
27:43This is probably the only case I know of where someone worked for such a long time without divulging what he was doing, without talking about the progress he had made. It's just unprecedented.
28:00Andrew was embarking on one of the most complex calculations in history.
28:06For the first two years he did nothing but immerse himself in the problem, trying to find a strategy which might work.
28:13So, it was now known that Tanyama Shimura implied Fermat's last theorem. What does Tanyama Shimura say? It says that all elliptic curves should be modular.
28:29Well, this was an old problem, been around for 20 years, and lots of people would try to solve it.
28:39Now, one way of looking at it is that you have all elliptic curves, and then you have the modular elliptic curves, and you want to prove that they're the same number of each.
28:47Now, of course, you're talking about infinite sets, so you can't just count them per se, but you can divide them into packets, and you can try to count each packet and see how things go.
28:55And this proves to be a very attractive idea for about 30 seconds, but you can't really get much further than that.
29:01And the big question in the subject was how you could possibly count, and in effect, Wiles introduced the correct technique.
29:07Andrew Wiles hoped to solve the problem of counting elliptic curves by converting them into something called Galois representations.
29:20Although no less complex than elliptic curves, they were easier to count.
29:24So it was Galois representations, not elliptic curves, that Andrew would now compare with modular forms.
29:33Now you might ask, and it's an obvious question, why can't you do this with elliptic curves and modular forms?
29:38Why couldn't you count elliptic curves, count modular forms, show that they're the same number?
29:45Well, the answer is people tried, and they never found a way of counting them.
29:49And this was why this is the key breakthrough, that I had found a way to count, not the original problem, but the modified problem.
29:57I found a way to count modular forms and Galois representations.
30:00This was only the first step, and already it had taken three years of Andrew's life.
30:11My wife's only known me while I've been working on Fairmother.
30:15I told her a few days after we got married.
30:20I decided that I really only had time for my problem and my family.
30:24When I was concentrating very hard, then I found that, with young children, that's the best possible way to relax.
30:35When you're talking to young children, they simply aren't interested in Fairmother, at least this age.
30:43They want to hear a children's story, and they're not going to let you do anything else.
30:46So, I'd found this wonderful counting mechanism, and I started thinking about this concrete problem in terms of Iwasawa theory.
30:57Iwasawa theory was the subject I'd studied as a graduate student, and in fact, with my advisor, John Coates, I'd used it to analyze elliptic curves.
31:08Iwasawa theory, Andrew hoped, would be the key to completing his counting strategy.
31:15Now, I tried to use Iwasawa theory in this context, but I ran into trouble.
31:29I seemed to be up against a wall.
31:39I just didn't seem to be able to get past it.
31:42Well, sometimes when I can't see what to do next, I often come here by the lake.
32:02Walking has a very good effect in that you're in this state of concentration, but at the same time, you're relaxing, you're allowing the subconscious to work on you.
32:18Andrew struggled for months using Iwasawa theory in an effort to create something called a class number formula.
32:36Without this critical formula, he would have nowhere left to go.
32:40So, at the end of the summer of 91, I was at a conference, and John Coates told me about a wonderful new paper of Matthias Flack, a student of his, in which he had tackled a class number formula.
32:53In fact, exactly the class number formula I needed.
32:57So, Flack, using ideas of Kali Vargin, had made a very significant first step in actually producing the class number formula.
33:10So, at that point, I thought, this is just what I need, this is tailor-made for the problem.
33:17I put aside completely the old approach I've been trying, and I devoted myself day and night to extending his result.
33:26Andrew was almost there, but this breakthrough was risky and complicated.
33:46After six years of secrecy, he needed to confide in someone.
33:50In January of 1993, Andrew came up to me one day at tea, asked me if I could come up to his office.
33:57There was something he wanted to talk to me about.
33:59I had no idea what this could be.
34:02I went up to his office, he closed the door, he said he thought he would be able to prove Tony Amishimura.
34:09I was just amazed, this was fantastic.
34:13It involved a kind of mathematics that Nick Katz is an expert in.
34:20I think another reason he asked me was that he was sure I would not tell other people.
34:28I would keep my mouth shut, which I did.
34:30Andrew Wiles and Nick Katz had been spending rather a lot of time huddled over a coffee table at the far end of the common room working on some problem or other.
34:45We never knew what it was.
34:47To avoid any more suspicion, Andrew decided to check his proof by disguising it in a series of lectures at Princeton, which Nick Katz could attend.
34:56Well, I explained at the beginning of the course that Flack had written this beautiful paper and I wanted to try to extend it to prove the full class number formula.
35:07The only thing I didn't explain was that proving the class number formula was most of the way to Fermat's last theorem.
35:18So this course was announced, it said calculations on elliptic curves, which could mean anything.
35:22Didn't mention Fermat, didn't mention Taniyama Shamura, there was no way in the world anyone could have guessed that it was about that if he didn't already know.
35:34None of the graduate students knew and in a few weeks they just drifted off because it's impossible to follow stuff if you don't know what it's for, pretty much.
35:41It's pretty hard even if you don't know what it's for, but after a few weeks I was the only guy in the audience.
35:53The lectures revealed no errors and still none of his colleagues suspected why Andrew was being so secretive.
36:00Maybe he's run out of ideas. That's why he's quiet, you never know why they're quiet.
36:11The proof was still missing a vital ingredient, but Andrew now felt confident it was time to tell one more person.
36:18So I called up Peter and asked him if I could come round and talk to him about something.
36:33I got a phone call from Andrew saying that he had something very important he wanted to chat to me about.
36:38And sure enough, he had some very exciting news.
36:44He said, I think you better sit down for this.
36:47He sat down and I said, I think I'm about to prove Fama's last theorem.
36:52I was flabbergasted, excited, disturbed.
36:57I mean, I remember that night finding it quite difficult to sleep.
37:00But there was still a problem.
37:14Late in the spring of 93, I was in this very awkward position that I thought I'd got most of the elliptic curves being modular.
37:23So that was nearly enough to be content to have Fama's last theorem.
37:28But there was this, these few families of elliptic curves that had escaped the net.
37:35And I was sitting here at my desk in May of 93, still wondering about this problem.
37:44And I was casually glancing at a paper of Barry Mazur's.
37:49And there was just one sentence which made a reference to actually what's the 19th century construction.
37:55And I just instantly realized that there was a trick that I could use, that I could switch from the families of elliptic curves I'd been using.
38:06I'd been studying them using the prime three.
38:08I could switch and study them using the prime five.
38:10It was, it looked more complicated but I could switch from these awkward curves that I couldn't prove were modular to a different set of curves which I'd already proved were modular and use that information to just go that one last step.
38:25And I just kept working out the details and the time went by and I forgot to go down to lunch and it got to about tea time and then I went down and Nada was very surprised that I'd arrived so late.
38:41And then she, I told her that I, I believed I'd solved Femi's last theorem.
38:50I was convinced that I had Femi in my hands and there was a conference in Cambridge organized by my advisor, John Coates.
39:07I thought that would be a wonderful place. It's my old hometown. I've been a graduate student there. It'd be a wonderful place to talk about it if I could get it in good shape.
39:20The name of the lectures that he announced was simply elliptic curves and modular forms. There was no mention of Femi's last theorem.
39:39Well, I was at this conference on L functions and elliptic curves. It was kind of a standard conference and all of the people were there.
39:46Didn't seem to be anything out of the ordinary until people started telling me that they'd been hearing weird rumors about Andrew Wallace's proposed series of lectures.
39:57I started talking to people and I got more and more precise information. I have no idea how it was spread.
40:03Not from me. Not from me.
40:06Whenever any piece of mathematical news had been in the air, Peter would say, oh, that's nothing. Wait until you hear the big news.
40:13There's something big going to break. Maybe some hints, yeah.
40:18People would ask me leading up to my lectures what exactly I was going to say.
40:24And I said, well, come to my lecture and see.
40:27It's a sort of very charged atmosphere. A lot of the major figures of the parathematical algebraic geometry were there.
40:39Richard Taylor, John Coates, Barry Mazur.
40:41Well, I'd never seen a lecture series of mathematics like that before.
40:46What was unique about those lectures were the glorious ideas, how many new ideas were presented, and the constancy of its dramatic build-up. It was suspenseful until the end.
40:59Well, it was this marvelous moment when we were coming close to a proof of Fairmont's Last Theorem. The tension had built up and there was only one possible punchline.
41:10So, after I'd explained the 3-5 switch on the backboard, I then just wrote up a statement of Fairmont's Last Theorem. I said I'd proved it. I said I think I'll stop there.
41:25The next day, what was totally unexpected was that we were deluged by inquiries from newspapers, journalists from all around the world.
41:43It was a wonderful feeling after seven years to have really solved my problem. I'd finally done it. Only later did it come out that there was a problem at the end.
42:13Now it was time for it to be refereed, which is to say for people appointed by the journal to go through and make sure that the thing was really correct.
42:34So, for two months, July and August, I literally did nothing but go through this manuscript line by line.
42:41And what this meant concretely was that essentially every day, sometimes twice a day, I would email Andrew with a question.
42:51I don't understand what you say. On this page, on this line, it seems to be wrong. I just don't understand.
42:57So, Nick was sending me emails and at the end of the summer, he sent one that seemed innocent at first and I tried to resolve it.
43:07And it's a little bit complicated, so he sends me a fax, but the fax doesn't seem to answer the question, so I email him back and I get another fax, which I'm still not satisfied with.
43:17And this, in fact, turned into the error that turned out to be a fundamental error and that we had completely missed when he was lecturing in the spring.
43:29So, that's where the problem was, in the method of Flach and Kollerwagen that I'd extended.
43:36So, once I realized that at the end of September, that there was really a problem with the way I'd made the construction, I spent the fall trying to think what kind of modifications could be made to the construction.
43:56So, there are lots of simple and rather natural modifications, any one of which might work.
44:02And every time he would try to fix it in one corner, some other difficulty would add up in another corner.
44:12It was like he was trying to put a carpet in a room where the carpet had more size than the room, but he could put it in any corner, and then when he rounded the other corners, it would pop up in this corner.
44:23And whether you could not put the carpet in the room was not something that he was able to decide.
44:28So, in September 93, when proof was running into problems, Nada said to me, the only thing I want for my birthday is the correct proof.
44:47My birthday is October 6, I had two or three weeks, and I failed to deliver.
44:54I think he externally appeared normal, but at this point he was keeping a secret from the world.
45:07And I think he must have been in fact pretty uncomfortable about it.
45:13Towards the end of November, it didn't seem to be working, I sent out an email message announcing that there was a problem with this part of the argument.
45:28Well, you know, we were behaving a little bit like Kremlinologists, nobody actually liked to come out and ask him how he's getting on with the proof.
45:39So, somebody would say, I saw Andrew this morning, did he smile?
45:46And, well, yes, but he didn't look too happy.
45:49The first seven years I'd worked on this problem, I loved every minute of it.
45:56However hard it had been, there'd been setbacks often, there'd been things that had seemed insurmountable,
46:04but it was a kind of private and very personal battle I was engaged in.
46:12And then, after there was a problem with it, doing mathematics in that kind of rather overexposed way is certainly not my style,
46:30and I have no wish to repeat it.
46:32After months of failure, Andrew was about to admit defeat.
47:01In desperation, he decided to ask for help, and a former student, Richard Taylor, came to Princeton.
47:13So, Richard and I spent three months at the beginning of 1994 trying to analyze all the possible modifications,
47:22and at the end of that period I was convinced that none of them was really going to give the answer.
47:31In September, I decided to go back and look one more time at the original structure of Flack and Kohlvergen to try and pinpoint exactly why it wasn't working.
47:53Try and formulate it precisely. One can never really do that in mathematics, but I just wanted to set my mind at rest,
48:00that it really couldn't be made to work.
48:03I was sitting here at this desk, it was a Monday morning, September 19th, and I was trying, convincing myself that it didn't work,
48:22just seeing exactly what the problem was, when suddenly, totally unexpectedly, I had this incredible revelation.
48:32I realized what was holding me up was exactly what would resolve the problem I'd have in my Iwasawa theory attempt three years earlier.
48:45It was, it was the most, the most important moment of my working life.
48:58It was so indescribably beautiful. It was so simple and so elegant.
49:19And I just stared in disbelief for 20 minutes. Then, during the day, I walked around the department,
49:29I'd keep coming back to my desk and looking to see it was still there, it was still there.
49:37Almost what seemed to be stopping the method of Flack and Kohlvergen was exactly what would make,
49:42horizontally with Iwasawa theory, my original approach to the problem from three years before,
49:47would make exactly that work. So, out of the ashes seem to rise the true answer to the problem.
50:02So, the first night I went back and slept on it, I checked through it again the next morning,
50:09and about 11 o'clock, I was satisfied and I went down and told my wife,
50:17I've got it, I think I've got it, I found it. And it was so unexpected,
50:23I think she thought I was talking about a children's toy or something.
50:27She said, got what? And I said, I've fixed my proof, I've got it.
50:31I think it will always stand as one of the high achievements of number theory.
50:50It was magnificent.
50:52It's not every day that you hear the proof of the century.
50:54Well, my first reaction was, I told you so.
51:06The Taniyama-Shimura conjecture is no longer a conjecture.
51:10And as a result, Fermat's last theorem has been proved.
51:13But is Andrew's proof the same as Fermat's?
51:19Fermat's proof was just too big to fit into this margin.
51:23Andrew's was 200 pages long. It's not the same proof.
51:28Fermat couldn't possibly have had this proof.
51:32It's a 20th century proof. There's no way this could have been done before the 20th century.
51:37I'm relieved that this result is now settled.
51:43But I'm sad in some ways because Fermat's last theorem has been responsible for so much.
51:50What will we find take its place?
51:55There's no other problem that will mean the same to me.
51:59I had this very rare privilege of being able to pursue
52:02in my adult life what had been my childhood dream.
52:11I know it's a rare privilege, but if one can do this,
52:16if one can really tackle something in adult life that means that much to you,
52:28it's more rewarding than anything I can imagine.
52:30One of the great things about this work is that it embraces the ideas of so many mathematicians.
52:49I've made a partial list.
52:51Klein, Frecke, Hurwitz, Hecke, Dirichlet, Derecken.
52:57Approved by Langlands and Tunnel.
52:59Deline, Rapoport, Ketz.
53:02Mazer's idea of using the deformation theory of Galois representations.
53:06Igusa, Eichler, Shimura, Tonyama.
53:10Fry's reduction. The list goes on and on.
53:12Block, Kato, Selmer, Fry, Fermat.
53:18There was another player in the Fermat game.
53:35She lived during the French Revolution and pretended to be a man in order to pursue her passion for mathematics.
53:41At NOVA's website, meet Sophie Germain at www.pbs.org.
53:48Educators can order this show for $19.95 plus shipping and handling by calling 1-800-255-9424.
54:06And, to learn more about how science can solve the mysteries of our world, ask about our many other NOVA videos.
54:17It's like Everest. It won't go away. It still stays there.
54:20Well, mathematicians just love a challenge.
54:23And this problem, this particular problem, just looks so simple.
54:26It just looked as if it had to have a solution.
54:29Andrew Wiles is probably one of the few people on Earth who had the audacity to dream that you can actually go and prove this conjecture.
54:36To see the enormous part, I would circle the sun to see.
54:40Ah good, look at the consequences, humans just what are real?
54:44I wish to see the moon a picture or find the moon a world.
54:50Oh wicked.
54:55I wish I was as if mother and my wife had reached to the moon!
54:59There was a flood inrill other patients, whereas nadal was in Greek century.
55:02NOVA is a production of WGBH Boston.
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55:26And by the Corporation for Public Broadcasting and viewers like you.
55:32This is PBS.
55:45To learn more about this subject, you can order For Ma's Enigma, the companion book to this program, by calling 1-800-255-9424.
55:56This hardcover edition is $23 plus shipping and handling.
56:02The Park Foundation is a production of WGBH.
56:04The Park Foundation is a production of WGBH.
56:06Coming up on NOVA, they are beasts of legend, feared by many, understood by few.
56:12Now, NOVA tracks them up close and captures behavior never before seen.
56:18Wild Wolves
56:20Next time on Mobile Masterpiece Theatre...
56:27It goes on forever.
56:29A gem precious enough to die for, or kill for.
56:33This is not a common case of thieving.
56:36Drive on!
56:37It's a diamond.
56:38And the name of it is the Moon Stone.
56:43Nooga
56:49You