The Last Theorem

THE THIRD POSTAMBLE

We felt it would be useful to give more details of what Fermat’s Last Theorem was all about, but we could not find an earlier place for this discussion that did not wound, almost fatally, the story’s narrative pace. So here it is at the end…and, if you are part of that large fraction of humanity who doesn’t know it all already, we do think you will find that it was worth waiting for.

The story of the most famous problem in mathematics began with a casual jotting by a seventeenth-century French attorney from Toulouse. The attorney’s name was Pierre de Fermat. Lawyering did not take up all of Fermat’s time, and so he dabbled in mathematics as an amateur—or, to give him his due, actually as a person with a solid claim to being called one of the greatest mathematicians of all time.

The name of that famous problem is Fermat’s Last Theorem.

One of that theorem’s greatest appeals is that it is not at all hard to understand. In fact, for most people coming to it for the first time, it is hard to believe that proving something so elementary that it can be demonstrated by counting on one’s fingers had defied all the world’s mathematicians for more than three centuries. In fact the problem’s origins go back a lot further than that, because it was Pythagoras himself, around five hundred B.C., who defined it in the words of the only mathematical theorem that has ever become a cliché:

“The square of the hypotenuse of a right triangle equals the sum of the squares of the opposite sides.”

For those of us who got as far as high school freshman math, we can visualize a right triangle and thus write the Pythagorean theorem as a² + b² = c².

Other mathematicians began investigating matters related to Pythagoras’s statement about as soon as Pythagoras stated it (that is what mathematicians do). One discovery was that there were many right triangles with whole-number sides that fit the equation. Such a triangle with sides of five units and twelve units, for instance, will have a hypotenuse measuring thirteen units…and, of course, 5² plus 12² does in fact equal 13². Some people looked at other possibilities. Was there, for example, any whole-number triangle with a similar relation to the cubes of the arms? That is, could a³ plus b³ ever equal c³? And what about fourth-power numbers, or indeed numbers with an exponent of any number other than two?

In the days before mechanical calculators, let alone electronic ones, people spent lifetimes squandering acres of paper with the calculations necessary to try to find the answers to such questions. So they did on this problem. No one found any answers. The amusing little equation worked for squares but not for any other exponent.

Then everyone stopped looking, because Fermat had stopped them with a single scrawled line. That charming little equation that worked for squares would never work for any other exponent, he said. Positively.

Now, most mathematicians would have published that statement in some mathematical journal. Fermat, however, was in some ways a rather odd duck, and that wasn’t his style. What he did was make a little note in the white space of a page of his copy of the ancient Greek mathematics book called Arithmetica. The note said:

“I have discovered a truly marvelous proof of this proposition which this margin is too narrow to contain.”

What made this offhand jotting important was that it contained the magic word “proof.”

A proof is powerful medicine for mathematicians. The requirement of a proof—that is, of a logical demonstration that a given statement must always and necessarily be true—is what distinguishes mathematicians from most “hard” scientists. Physicists, for instance, have it pretty easy. If a physicist splatters a bunch of high-velocity protons onto an aluminum target ten or a hundred times and always gets the same mix of other particles flying out, he is allowed to assume that some other physicist doing the same experiment somewhere else will always get the same selection of particles.

The mathematician is allowed no such ease. His theorems aren’t statistical. They must be definitive. No mathematician is allowed to say that any mathematical statement is “true” until he has, with impeccable and unarguable logic, constructed a proof that shows that this must always be the case—perhaps by showing that if it were not, it would lead to an obvious and absurd contradiction.

So then the real search began. Now what the mathematicians were looking for was the proof that Fermat had claimed to possess. Many of the greatest mathematicians—Euler, Goldbach, Dirichlet, Sophie Germain—did their best to find that elusive proof. So did lesser names by the hundreds. From time to time some weary one of their number would leap to his feet with a cry of joy and a claim to have found the solution. Such alleged “proofs” turned up in the hundreds; there were a thousand of them in one four-year stretch of the early twentieth century alone.

But they were all quickly slain by other mathematicians who found the writers had made fundamental mistakes in fact or in logic. It began to seem to the mathematical world that great Fermat had stumbled and that no proof of his scribble would ever be found.

In that conclusion, however, they were not entirely right.

A true and final proof of Fermat’s theorem came at last at almost the end of the twentieth century. It happened in the years 1993 to 1995, when a British mathematician named Andrew Wiles, working at Princeton University in the United States, published a final, complete, error-free, and definitive proof of Fermat’s 350-odd-year-old conjecture. The problem had been solved.

Hardly anyone, however, was satisfied.

In the first place, Wiles’s proof was ridiculously long—one hundred fifty densely written pages. Worse still, there were parts of it that no human being could read—and thereby confirm that they were error-free. Only a computer program could check them. Worst of all, Wiles’s proof could not have been the one that Fermat claimed, because it relied on proofs and procedures that had not been known to Fermat or anyone else anywhere near that time. So, many great mathematicians refused to accept it….

Including, as we have just seen, one truly superb, if fictitious, one. We are speaking of one whose home was far from Fermat’s in both time and space, namely the one named Ranjit Subramanian, whom this book has been about.