I don’t know who you are and what you know already. If you would be a research level mathematician with a sound knowledge of algebra, algebraic geometry. Fermat’s Last Theorem was until recently the most famous unsolved problem in mathematics. In the midth century Pierre de Fermat wrote that no value of n. On June 23, , Andrew Wiles wrote on a blackboard, before an audience A proof by Fermat has never been found, and the problem remained open.
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Andrew Wiles Andrew Wiles devoted much of his entire career to proving Fermat’s Last Theorem, the world’s most famous mathematical problem. Inhe made front-page headlines when he announced a proof of the problem, germat this was not the end of the story; an error in his calculation jeopardized his life’s work. Andrew Wiles spoke to NOVA and described how he came to terms with the mistake, and eventually went on to achieve his life’s ambition.
Fermat’s Last Theorem proof secures mathematics’ top prize for Sir Andrew Wiles
Many great scientific discoveries are the result of obsession, but in your case that obsession has held you since you were a child.
I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school. I’d take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem—Fermat’s Last Theorem.
This problem had been unsolved by mathematicians for years. It looked so simple, and yet all the great mathematicians in history couldn’t solve it. Here was a problem, that I, a ten year old, could understand and I knew from that moment that I would never let it go. I had to solve it.
Who was Fermat and what was his Last Theorem?
Wiles’s proof of Fermat’s Last Theorem
Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras’ equation. Pythagoras’ equation gives you: Fermat then considered the cubed andreww of this equation: He claimed that there were none.
Fremat fact, he claimed that for the general family of equations: That’s Fermat’s Last Theorem. So Fermat said because he could not find any solutions to this equation, then there were no solutions? He did more than that. Just because we can’t find a solution it doesn’t mean that there isn’t one. Mathematicians aren’t satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity.
And to do that we need a proof. Fermat said he had a proof. Unfortunately, all he ever wrote down was: What do you mean by a proof? In a mathematical proof you have a line of reasoning consisting of many, many steps, that are almost self-evident. If the proof we write down is really rigorous, then nobody can ever prove it wrong. There are proofs that date back to the Greeks that are still valid today.
So the challenge was to rediscover Fermat’s proof of the Last Theorem. Why did it become so famous? Well, some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they’re extremely hard to solve. There’s no reason why these problems shouldn’t be easy, and yet they turn out to be extremely intricate.
The Last Theorem is the most beautiful example of this. But finding a proof has no applications in the real world; it is a purely abstract question. So why have people put so much effort into finding a proof? Pure mathematicians just love to try unsolved problems—they love a challenge. And as time passed and no proof was found, it became a real challenge.
I’ve read letters andfew the early 19th century which said that it was an embarrassment to mathematics that the Last Theorem had not been solved. And of course, it’s very special because Fermat said that he had a proof. How did you begin looking for the proof? In my early teens I tried to tackle the problem as I thought Fermat might have tried it.
I reckoned that he wouldn’t have known much more math than I knew as a teenager. Then when I reached college, I realized that fsrmat people andreww thought about the problem during the 18th and 19th centuries and so I studied those methods.
But I still wasn’t getting anywhere. Then when I became a researcher, I decided that I should put the problem aside. It’s not that I forgot about it—it was always there—but I realized that the only techniques we had to tackle it had been around for years.
NOVA Online | The Proof | Solving Fermat: Andrew Wiles
It didn’t seem that these techniques were really getting to the root of the ferat. The problem with working on Fermat was that you could spend years getting nowhere.
It’s fine to work on any problem, so long as it generates interesting mathematics along the way—even if you don’t solve it at the end of the day.
The definition of a good mathematical problem is the mathematics it generates rather than the problem itself. It seems that the Last Theorem was zndrew impossible, and that mathematicians could not risk wasting getting nowhere. But then in everything changed.
Can you remember how you reacted to this news? 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 fsrmat told me that Ken Ribet had proved a link between Taniyama-Shimura and Fermat’s Last Theorem.
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. So, because Taniyama-Shimura was a modern problem, this meant that working on it, and by implication trying to prove Fermat’s Last Theorem, was respectable. Nobody had any idea how to approach Taniyama-Shimura but at least prolf 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.
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. I realized that anything to do with Fermat’s Last Theorem generates too much interest. You can’t really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed. But presumably you told your wife what you were doing?
My wife’s only known me while I’ve been working on Fermat. 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 prpof significance it had for mathematicians, that it had been such a thorn in our flesh for so many years.
On a day-to-day basis, how did you go about constructing your proof? 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.
Fermat’s Last Theorem — from Wolfram MathWorld
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 sndrew new; it’s a mystery where that comes from.
I wules this problem around in my head basically the whole time. I would wake up with it first fernat 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 fernat a story and they’re not going to let you do anything else.
Usually people work in groups and feermat each other for support. What did you do when you hit a brick wall? When I got stuck and I didn’t know what to do next, I would go out for a walk. I’d often walk down by the lake. Walking has a very good effect adrew 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. So for seven years you’re pursuing this proof. Presumably there are periods of self-doubt mixed with the periods of success.
Perhaps Prood 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.