The Last Theorem - Arthur Charles Clarke [162]
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,