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But then in everything changed. Can you remember how you reacted to this news? AW: It was one evening at the end of the summer of when I was sipping iced tea at the house of a friend. Casually in the middle of a conversation this friend told me that Ken Ribet had proved a link between Taniyama-Shimura and Fermat's Last Theorem. I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat's Last Theorem all I had to do was to prove the Taniyama-Shimura conjecture.

It meant that my childhood dream was now a respectable thing to work on. I just knew that I could never let that go. AW: Yes.

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Nobody had any idea how to approach Taniyama-Shimura but at least it was mainstream mathematics. I could try and prove results, which, even if they didn't get the whole thing, would be worthwhile mathematics. So the romance of Fermat, which had held me all my life, was now combined with a problem that was professionally acceptable.

NOVA: At this point you decided to work in complete isolation. You told nobody that you were embarking on a proof of Fermat's Last Theorem. Why was that? You can't really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed. NOVA: But presumably you told your wife what you were doing? AW: My wife's only known me while I've been working on Fermat.

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I told her on our honeymoon, just a few days after we got married. My wife had heard of Fermat's Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years. NOVA: On a day-to-day basis, how did you go about constructing your proof?

AW: I used to come up to my study, and start trying to find patterns. I tried doing calculations which explain some little piece of mathematics. I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.

Sometimes that would involve going and looking it up in a book to see how it's done there. Sometimes it was a question of modifying things a bit, doing a little extra calculation. And sometimes I realized that nothing that had ever been done before was any use at all. Then I just had to find something completely new; it's a mystery where that comes from. I carried this problem around in my head basically the whole time.

The Romance of Fermat’s Last Theorem

I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction, I would have the same thing going round and round in my mind. The only way I could relax was when I was with my children. Young children simply aren't interested in Fermat. They just want to hear a story and they're not going to let you do anything else. NOVA: Usually people work in groups and use each other for support.

What did you do when you hit a brick wall?

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I'd often walk down by the lake. Walking has a very good effect in that you're in this state of relaxation, but at the same time you're allowing the sub-conscious to work on you. And often if you have one particular thing buzzing in your mind then you don't need anything to write with or any desk. I'd always have a pencil and paper ready and, if I really had an idea, I'd sit down at a bench and I'd start scribbling away.

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NOVA: So for seven years you're pursuing this proof. Presumably there are periods of self-doubt mixed with the periods of success. AW: Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark.

You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that proceed them.

NOVA: And during those seven years, you could never be sure of achieving a complete proof.

AW: I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal. It could be that the methods needed to take the next step may simply be beyond present day mathematics. Perhaps the methods I needed to complete the proof would not be invented for a hundred years.

So even if I was on the right track, I could be living in the wrong century. NOVA: Then eventually in , you made the crucial breakthrough. AW: Yes, it was one morning in late May. My wife, Nada, was out with the children and I was sitting at my desk thinking about the last stage of the proof. I was casually looking at a research paper and there was one sentence that just caught my attention. It mentioned a 19th-century construction, and I suddenly realized that I should be able to use that to complete the proof.

I went on into the afternoon and I forgot to go down for lunch, and by about three or four o'clock, I was really convinced that this would solve the last remaining problem. It got to about tea time and I went downstairs and Nada was very surprised that I'd arrived so late. What was the error? AW: It was an error in a crucial part of the argument, but it was something so subtle that I'd missed it completely until that point. The error is so abstract that it can't really be described in simple terms.

Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail. NOVA: Eventually, after a year of work, and after inviting the Cambridge mathematician Richard Taylor to work with you on the error, you managed to repair the proof.

The question that everybody asks is this; is your proof the same as Fermat's?

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AW: There's no chance of that. Fermat couldn't possibly have had this proof. It's pages long. It's a 20th-century proof. It couldn't have been done in the 19th century, let alone the 17th century. The techniques used in this proof just weren't around in Fermat's time. AW: I don't believe Fermat had a proof.