Solving Fermat: Andrew Wiles
Andrew Wiles devoted much of his entire career to proving Fermat's Last Theorem,
the world's most famous mathematical problem. In 1993, he made front-page
headlines when he announced a proof of the problem, but 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.
NOVA: Many great scientific discoveries are the result of obsession, but in
your case that obsession has held you since you were a child.
ANDREW WILES: 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 300 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.
NOVA: Who was Fermat and what was his Last Theorem?
AW: 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:|
x2 + y2 = z2
You can ask, what are
the whole number solutions to this equation, and you can see that:
32 + 42 = 52
52 + 122 = 132
And if you go on looking then you find more and more such
solutions. Fermat then considered the cubed version of this equation:
x3 + y3 = z3
He raised the question: can you find solutions to the cubed equation? He
claimed that there were none. In fact, he claimed that for the general family
xn + yn = zn where n is bigger than 2
it is impossible to find
a solution. That's Fermat's Last Theorem.
NOVA: So Fermat said because he could not find any solutions to this equation,
then there were no solutions?
AW: 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:
"I have a truly marvelous demonstration of this proposition which this margin
is too narrow to contain."|
NOVA: What do you mean by a proof?
AW: 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.
NOVA: So the challenge was to rediscover Fermat's proof of the Last Theorem.
Why did it become so famous?
AW: 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.
NOVA: 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
AW: 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 in 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.
NOVA: How did you begin looking for the proof?
AW: 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 many people
had 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 130 years. It didn't seem
that these techniques were really getting to the root of the problem. 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.|
NOVA: It seems that the Last Theorem was considered impossible, and that
mathematicians could not risk wasting getting nowhere. But then in 1986
everything changed. A breakthrough by Ken Ribet at the University of California
at Berkeley linked Fermat's Last Theorem to another unsolved problem, the
Taniyama-Shimura conjecture. Can you remember how you reacted to this news?
AW: It was one evening at the end of the summer of 1986 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.|
NOVA: 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
AW: Yes. 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
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?
AW: 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.
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. 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. 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
NOVA: Usually people work in groups and use each other for support. What did
you do when you hit a brick wall?
AW: 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 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.|
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
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 1993, 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. Then I told her I'd solved Fermat's
NOVA: The New York Times exclaimed "At Last Shout of 'Eureka!' in Age-Old Math
Mystery," but unknown to them, and to you, there was an error in your proof.
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?
AW: There's no chance of that. Fermat couldn't possibly have had this proof.
It's 150 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.
NOVA: So Fermat's original proof is still out there somewhere.
AW: I don't believe Fermat had a proof. I think he fooled himself into thinking
he had a proof. But what has made this problem special for amateurs is that
there's a tiny possibility that there does exist an elegant 17th-century
NOVA: So some mathematicians might continue to look for the original proof.
What will you do next?
AW: There's no problem that will mean the same to me. Fermat was my childhood
passion. There's nothing to replace it. I'll try other problems. I'm sure that
some of them will be very hard and I'll have a sense of achievement again, but
nothing will mean the same to me. There's no other problem in mathematics that
could hold me the way that this one did. There is a sense of melancholy. We've
lost something that's been with us for so long, and something that drew a lot
of us into mathematics. But perhaps that's always the way with math problems,
and we just have to find new ones to capture our attention. People have told me
I've taken away their problem—can't I give them something else? I feel some
sense of responsibility. I hope that seeing the excitement of solving this
problem will make young mathematicians realize that there are lots and lots of
other problems in mathematics which are going to be just as challenging in the
NOVA: What is the main challenge now?
AW: The greatest problem for mathematicians now is probably the Riemann
Hypothesis. But it's not a problem that can be simply stated.
NOVA: And is there any one particular thought that remains with you now that
Fermat's Last Theorem has been laid to rest?
AW: Certainly one thing that I've learned is that it is important to pick a
problem based on how much you care about it. However impenetrable it seems, if
you don't try it, then you can never do it. Always try the problem that matters
most to you. I had this rare privilege of being able to pursue in my adult
life, what had been my childhood dream. I know it's a rare privilege, but if
one can really tackle something in adult life that means that much to you, then
it's more rewarding than anything I can imagine.
NOVA: And now that journey is over, there must be a certain sadness?
AW: There is a certain sense of sadness, but at the same time there is this
tremendous sense of achievement. There's also a sense of freedom. I was so
obsessed by this problem that I was thinking about it all the time—when I
woke up in the morning, when I went to sleep at night—and that went on for
eight years. That's a long time to think about one thing. That particular
odyssey is now over. My mind is now at rest.
Andrew Wiles |
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© | Updated November 2000