Marginal at best

For many years, Edmund Landau, a German mathematician, had a form letter that looked like this: "Dear Sir/Madam: Your proof of Fermat's Last Theorem has been received. The first mistake is on page _____, line _____." Landau would assign the job of filling in those blanks to one of his students. They must have been busy, because crank "proofs" of Fermat were something of a cottage industry; if I recall right, Orissa was a minor breeding ground for them.

But Landau's form letters are likely gathering dust today. Not just because the man passed on a long time ago, but because Fermat's Last Theorem (FLT) has actually been proved. Undoubtedly the best-known unsolved problem in mathematics for over 350 years, this little conundrum was finally solved in 1993 by an English mathematician called Andrew Wiles. This marginal essay is too small to contain the mathematics of Wiles' proof, and I wouldn't understand it anyway. But many crank talents now like to think they do: the cottage industry has turned to claims of disproving Wiles. Time to resurrect and change those form letters, I think.

FLT's appeal, the reason it was such a target for amateur mathematicians, is that the problem is so easily understood. You don't need any mathematical training to follow it. You need only know about raising a number to a power -- multiplying it by itself a certain number of times. For example, 2³ (2 raised to the power of 3) = 2 x 2 x 2 = 8. Or, 5⁴ (5 raised to the power of 4) = 5 x 5 x 5 x 5 = 625.

Take this equation:

x² + y² = z²
(This should be familiar to those who remember a certain Shri Pythagoras and the properties of right-angled triangles). There are many trios of numbers that you can use as x, y, and z here. For example, 3² + 4² = 9 + 16 = 25 = 5², or 5² + 12² = 25 + 144 = 169 = 13² .

But what if you changed that power n to 3, or more? Now you're in FLT territory, and you bump into something startling. Take this equation:

xⁿ + yⁿ = zⁿ
FLT says that if n is greater than 2, there are no numbers that will fit x, y and z. In other words, you cannot find three numbers such that if you add the nth powers of two of them, you will get the nth power of the third -- if n is greater than 2.

Reading about this theorem in 1637, Pierre de Fermat made the most famous margin note in mathematics: "I have discovered a truly remarkable proof which this margin is too small to contain."

But Fermat never published his proof. The few scribbled words have tormented mathematicians, eminent and otherwise, ever since. Nobody could prove the theorem. It was such an intractable problem that mathematicians began to suspect Fermat had not actually found a proof. Perhaps he simply tried his hand at it for a while, found that standard methods were not working, and went on to other things. Perhaps he thought he was close enough to a proof that it didn't matter. Perhaps it was all an elaborate Fermat-ian joke.

Mathematicians who have tried to tackle FLT -- nearly every eminent one of the last three centuries -- have proved various related theorems. Each seemed like a step towards the Holy Grail. But with one exception, Wiles' solution was curious in that it didn't use any of these intermediate steps.

The exception was the work of two 19th Century German mathematicians, Kummer and Dirichlet, a cornerstone of modern algebra and the foundation of Wiles' proof. But the final step in the saga actually had its origins in the work of Yutaka Taniyama. In 1955, he was examining the properties of elliptic curves. With two others, he suggested a specific property of these curves, what's called the Shimura-Taniyama-Weil Conjecture (STW). In 1986, STW was shown to have connections with FLT: if STW was proved for a particular set of curves, FLT would also be proved.

(This connection showed that FLT is not some trivial curiosity of number theory, but has a lot to do with fundamental properties of curves and space).

It was left to Wiles to bridge the gap. In 1993, he proved STW for those particular curves, and so solved mathematics' greatest puzzle. There were some gaps in his work, but over the next couple of years, he took care of those. His eventual proof, like Fermat's, was also a little longer than a margin could contain: it actually filled an entire 130-page issue of the journal Annals of Mathematics.

And it's likely he did not dispatch his proof to Edmund Landau.

Wiles presented his results in a series of three lectures at Cambridge University in June 1993. At the end of the final lecture, he wrote Fermat's Last Theorem on the board, said "I will stop here," and sat down.

Boy, I'd have loved to have been there at that precise moment.