Fermat's Last Theorem

by Dino Surendran

This problem was written down around 1637 by Pierre de Fermat, a French lawyer in Toulouse who was also a prominent amateur mathematician. He was reading a textbook when a thought occurred to him. He decided to write it down before he forgot it --- and the nearest piece of paper was the margin of the said textbook:

``On the other hand it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or in general, for any number which is a power greater than the second to be written as a sum of two like powers. I have a truly marvellous demonstration of this proposition which this margin is too large to contain.''

Now, it is suspected that he later found that his proof was incorrect, since he only ever publicly stated solutions for third and fourth powers. But of course he didn't bother to correct this private note, and after his death his son published a new edition of the textbook with all his father's scribbles - including this intriguing footnote. Now every other theorem (lots!) Fermat claimed to have proved turned out to be true - but this problem stubbornly resisted all attempts by others to solve it and thus came to be known as Fermat's Last Theorem (FLT).

Interest in FLT rocketed when a German doctor and amateur mathematician called Paul Wolfskehl offered a huge cash prize in 1908 for its solution. Many people, mostly amateur mathematicians, sent in their potential solutions to their nearest universities. Over a thousand proofs were received in this period, and mail got so heavy that some institutions resorted to printing form sheets saying "Thank you for your `proof' of FLT. The first mistake is on page ___''! The hyperinflation of the 1920s considerably reduced the value of the prize but it was still worth US $50 000 when Wiles collected it at a grand ceremony in June 1997 in the Great Hall of Gottingen University.

To take a minor detour, why did Mr. Wolfskehl place so much importance on FLT? The reason is rather touching. When he was a young man he decided he was going to commit suicide. He planned the event diligently for midnight on some day. As the hour approached he got bored and picked up a copy of a maths book to kill time. He found a chapter on FLT and various attempts to prove it. He became so engrossed in the article that when he finally looked up from the book the hour was passed. He decided not to kill himself and went on to lead a productive life. He felt he owed his life to FLT, hence the reward.

Now, with such a glittering prize available, why did these mercenary attempts fail? Because the problem was actually hideously difficult. The legendary mathematician David Hilbert said of it: ``Before beginning (to prove it) I should put in three years of intensive study and I haven't that much time to squander on a probable failure.''

But progress was made, notably by the Japanese mathematicians Yutaka Taniyama (who killed himself in 1958) and Goro Shimura (who's a professor at Princeton University). Except they didn't know at the time that they were doing anything related to FLT. What they did was MUCH MORE important, in that it linked two apparently unrelated and farflung topics in number theory, elliptic equations and modular forms, in a powerful manner. People at first didn't think much of their idea but as evidence (though no proof) supporting it grew, it became known as the Taniyama-Shimura-Weil Conjecture, or TSWC. Andre Weil was a well-known French mathematician who worked with Shimura on the problem after Taniyama's death.

Fast forward slowly (!) to 1986, to two mathematicians having a cup of coffee at a café. One is Ken Ribet, a prof at the University of Berkeley, the other is Barry Mazur, a prof at Harvard. We've recreated, with little attempt at detailed historical accuracy, the following conversation between them:

Ken: ... and that was why X failed to get his tenureship.
Barry: Too bad. To change the subject though, how's your research going?
Ken: Research smesearch. Search search search and search again, that's why it's called re-search. You know the feeling when you're so close to something but you can't quite get there?
Barry: Yes, I've noticed the feeling, especially near pretty ladies.
Ken: Dolt. I was referring to my work on Frey's problem.
Barry: Frey Frey Frey... oh you mean Gerhard? That German guy who almost showed two years ago that TSWC implies FLT?
Ken: Yup. Brilliant piece of work he'd done. But as you say, he didn't quite get there --- there's one step of logic that still eluded him. And the rest of us. I've been trying to fix the gap, but all I've managed to do is a very special case blah blah blah (insert mathematical jargon here)...
Barry: But don't you see? You've already done it! You've completed what Frey started! All you have to do is moreblah meblah stillmoreblah...

Ken gazed intently at his coffee. This was certainly a dream, which meant his cup should disappear or something. But it remained - as did the proof upon a couple of hours of checking when he got home. He had indeed shown that anyone who proved TSWC would have proved FLT!!

Enter Andrew Wiles. When he saw that a new mode of attack (a hideously difficult one, but one in the mainstream of mathematics --- so that he wouldn't be throwing away his career even if he failed) was available to deal with FLT, he got excited. He felt he had to try and prove TSWC --- even though many intelligent people before him had tried and failed. He knew it would take possibly ten years, if he was lucky. But he had dreamed, since he first read about FLT as a boy of ten, of proving it and he dared to do what so many of us chicken out from --- making his dream come true.

His plan of attack: ``Learn everything there is to know about areas relating to TSWC. Play with existing techniques, practice practice practice. (This part alone took one and a half years.) Don't worry about what others are doing, don't tell them what you're doing. This is between you and maths. With only brain, pen and paper in between.''

Mathematicians don't usually work like this. In today's high pressure world, it's a risk they cannot usually take. But many breakthroughs in the past have occurred in this manner, made by brilliant AND diligent people working in isolation: Einstein on relativity, Gödel on logic, Taniyama and Shimura on TSWC. It's a hard thing to do and only a fraction of those who start on such lonely paths succeed. The rest die unknown, sometimes in poverty.

Not that Wiles cut himself off totally from the world - he kept his post at Princeton, and he got married at about this time. His wife Nada was the only person he told (on their honeymoon!) about his obsession. He worked very hard - often in his attic, which was an excellent place for peace and quiet. He remained quite human though, remarking later that ``the only way I could relax was when I was with my children. Young children aren't interested in Fermat, they just want to hear a story and they aren't going to let you do anything else.''

After seven solid years of work, Wiles announced his (200 page!) proof in June 1993, to much applause and media attention. He was so famous that one international fashion chain asked him to endorse a new line of men's clothing! (We have no idea if he accepted the offer.)

But the proof still had to be officially checked by other mathematicians. And during this `refereeing' process, a gap was discovered in the proof. Wiles was devastated. For despite frantic efforts, he could not set it right. Rumours began to spread that something was wrong, very wrong. One impatient mathematician queried about the reported gap in the proof: ``Does gap mean crack, fissure, crevasse, chasm or abyss?'' Wiles disappeared again. His colleagues at Princeton desperately wanted to ask him how things were going, but couldn't. Instead they gazed at his face every day to see if he was smiling or not!

Then, against all odds, his work with a former student (and by then a Cambridge professor) Richard Taylor led to a result in October 1994 that fixed the proof. This time, it was correct ALL the way.

Articles in this issue related to Fermat's Last Torture:

• Beal's Problem - the new FLT?
• Catalan's Conjecture - an alternative new FLT?
• A connection between FLT and our Math Department here at UZ

A very interesting site is Jeremy Teitelbaum's FLT Poetry Challenge, with such gems as this one due to Jonathan Matte: