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writing for eternity fermat’s last theorem

Pierre de Fermat was not a practical joker. The son of a wealthy leather merchant in southern France, he earned a law degree at the University of Orleans in 1631, bought a seat in the parlement (supreme court) at Toulouse, and became a member of the nobility. From the evidence of his letters, he was a shy, taciturn man who disliked controversy.

But Fermat had one unusual characteristic: He loved mathematics. In an era when mathematicians were starting to reach across national boundaries and turn their subject into an international enterprise, he achieved worldwide fame that lasted long after his death. By a curious twist of fate, his most lasting legacy was a problem that he almost certainly did not solve. That problem, called Fermat’s Last Theorem, unintentionally became the greatest practical joke in mathematical history—a deceptively simple statement that defied all efforts at proof for more than 350 years.

To the best of our knowledge, Fermat was self-taught. However, during his student days he formed friendships with a small circle of people who were interested in mathematics, and this apparently stimulated him to start doing his own research. One of his friends moved to Paris in 1636 to work in the royal library, and brought the work of this previously unknown provincial mathematician, Fermat, to the attention of Father Marin Mersenne.

The numbers x, y, z, and n are positive integers, and n is greater than 2. In contrast to the previous equations I have discussed, Fermat’s Last Theorem states that this equation has no solutions.

In an era before scientific academies and scientific journals, when most universities did not even have a professor of mathematics, Mersenne was the focal point of mathematics in France. He held regular meetings at his convent and kept in touch with nearly every mathematician in Europe. If you wanted to publicize a new discovery, you would send it to Mersenne. The rest of the world would find out soon enough.

Fermat himself never visited Paris, never ventured out of the south of France, and met Mersenne only one time, in 1646. He adamantly refused to have anything published under his own name. Nevertheless, his results became known everywhere, thanks to Mersenne, and other mathematicians in France and abroad avidly desired to learn his methods.

Yet Fermat was very close-lipped. His normal modus operandi was to send his discoveries as problems to other mathematicians, often artfully concealed so that the true nature of his discovery would not be apparent to the recipient unless they had been working on similar problems themselves. In this way Fermat could ascertain whether he had found something new, without giving away what it was.

Of course this sort of challenge both tantalized and annoyed other mathematicians. René Descartes called Fermat a “braggart,” and Bernard Frénicle de Bessy accused him of posing impossible problems. Fermat seems to have been torn between his desire for recognition and a nearly pathological fear of revealing too many of his secrets. On the one hand, he was fond of quoting a motto of Sir Francis Bacon: Multi pertranseant ut augeatur scientia (“Many must pass by in order that knowledge may grow”). On the other hand, by his reluctance to publish, Fermat made many parts of his own work inaccessible to “passersby.”

FERMAT WAS THE FIRST modern European mathematician to take an active interest in number theory, the study of equations with integer solutions. An ongoing theme in his work was Pythagorean triples: in other words, whole numbers a, b, and c such that a² + b² = c². As Fermat knew from studying a book that had been recently translated from the ancient Greek, Diophantus’ Arithmetica, the Greeks had a general method for solving this equation. Fermat came up with innumerable variations on the theme: finding two Pythagorean triples with the same hypotenuse c; Pythagorean triples whose areas were square numbers, or twice a square, or such that the sum of the legs a + b was square. He was able to resolve all of these problems to his satisfaction, even on occasion proving that there was no solution. (For example, no Pythagorean triangle has an area that is a perfect square.)

One day, probably between 1636 and 1640, he came up with another variation: Could a cube be written as a sum of two cubes? More generally, did the equation xⁿ + yⁿ = zⁿ ever have whole-number solutions if the exponent n was greater than 2? In the margin of his personal copy of Diophantus, Fermat wrote: “No cube can be split into two cubes, nor any biquadrate [fourth power] into biquadrates, nor generally any power beyond the second into two of the same kind. For this I have discovered a truly wonderful proof, but the margin is too small to contain it.” This handwritten note, which Fermat never intended anyone to see, became one of the most famous quotes in mathematical history. As number theorist André Weil has written, “How could he have guessed that he was writing for eternity?”

After Fermat died, his son Samuel collected and published his writings, including the copy of Diophantus with all of Fermat’s marginal notes. In the 1700s, the Swiss mathematician Leonhard Euler took as a personal challenge to (re)-prove all of Fermat’s results in number theory. The statement about splitting up powers into like powers was the only one that eluded him. He did prove that the equations x³ + y³ = z³ and x⁴ + y⁴ = z⁴ have no integer solutions, but he despaired of finding a general method for all n.

Fermat’s innocent marginal note became known as “Fermat’s Last Theorem.” Technically, of course, it was not a theorem (i.e., a proven fact), but a conjecture. In 1825, Peter Gustav Lejeune Dirichlet proved that there are no whole-number solutions if n = 5. In 1839, Gabriel Lamé proved likewise for n = 7. By 1857, Ernst Kummer had proved it for all exponents n up to 100. Even though progress seemed agonizingly slow, the efforts to prove Fermat’s Last Theorem were opening up new areas of mathematics, today called algebraic number theory.

In the twentieth century, Fermat’s Last Theorem continued to spawn new mathematics, like the goose that laid golden eggs. In the early 1980s a German mathematician, Gerhard Frey, realized that any putative solution to Fermat’s equation, aⁿ + bⁿ = cⁿ, could be used to construct an auxiliary curve, given by the equation y² = x(x – aⁿ)(x + bⁿ), which struck Frey as a highly bizarre specimen. It was so bizarre, Frey argued, that it would violate another unproven conjecture in number theory, called the Taniyama–Shimura conjecture. The evidence was circumstantial at first, but subsequently an American mathematician, Kenneth Ribet, proved that Frey was right—if the Taniyama–Shimura conjecture was true, so was Fermat’s Last Theorem.

Frey’s idea was forehead-smackingly clever. He turned the variables in Fermat’s equation into coefficients of a different equation. It’s like the reversal of the foreground and background in a picture by M.C. Escher. Even so, it was very far from obvious that Frey’s and Ribet’s work represented any kind of breakthrough. They had only exchanged one seemingly unattainable goal for another. In effect, Frey and Ribet said: You want to climb Mount Everest? It’s easy. Just grow wings.

Only one person in the world actually believed that he could prove the Taniyama–Shimura conjecture: Andrew Wiles. And he did it more or less by “growing wings.” Actually, he built an airplane. Over a seven-year period, working alone in his attic, he linked together three of the most difficult, abstract, powerful theories of twentieth-century mathematics—the theories of L-functions, modular forms, and Galois representations—into a smoothly functioning machine. One might compare his proof to the Apollo missions to the Moon, which combined (at least) three independent technologies: rocketry, computing, and communications. None of these technologies were developed with a Moon mission in mind. If any one of the three had been missing, the Moon missions would have been inconceivable. Yet they did come together, at just the right time, to conquer a famous “unsolved problem” (How can humans fly to the Moon?). Coincidentally, like Fermat’s Last Theorem, that problem had been around for just about 350 years.

Above Woodcut by Maurits Cornelius Escher, 1938, Sky and Water I, an example of reversal of the foreground and background

WILES ANNOUNCED HIS PROOF of Fermat’s Last Theorem in 1993. Unlike Fermat, Wiles submitted his proof for publication, in 1994. In the three-and-a-half centuries between Fermat and Wiles, mathematicians had learned their lesson: a “theorem” without a published proof is no theorem at all. In fact, as Wiles wrote up his proof in 1993, he discovered a gap that took him a year (plus the assistance of a student, Richard Taylor) to plug. Perhaps, if Fermat had taken the trouble to write down his proof, he would have discovered a gap as well.

And this brings us to an inescapable question: Did Fermat actually find a correct proof? The answer of any competent number theorist would be a resounding no. According to André Weil, we can be certain that Fermat had a proof for the n = 4 case, and we may plausibly believe that he found something like Euler’s proof for the n = 3 case. Both of these cases were solvable with Fermat’s “technology.” But beginning with n = 5, the problem changes very significantly. The case n = 5 required the nineteenth-century machinery of complex numbers and algebraic number fields. And, as I have described, Wiles’ proof of the general case required top-of-the-line twentieth-century concepts that Fermat could never have dreamed of.

To the mathematical argument Weil adds a psychological one. Fermat repeatedly bragged about the n = 3 and n = 4 cases and posed them as challenges to other mathematicians (including poor Frénicle). But he never mentioned the general case, n = 5 and higher, in any of his letters. Why such restraint? Most likely, Weil argues, because Fermat had realized that his “truly wonderful proof” did not work in those cases. Every mathematician has had days like this. You think you have a great insight, but then you go out for a walk, or you come back to the problem the next day, and you realize that your great idea has a flaw. Sometimes you can go back and fix it. And sometimes you can’t.

Weil’s mathematical and psychological arguments are compelling. However, I would like to give the last word to a class of high-school students I taught in 1990, three years before Wiles announced his proof. On the last day of the course, a group of them performed a skit based on Fermat’s life. As the curtain came down, they chanted in unison:

“Fermat! Fermat! He’s our man! If he can’t prove it, no one can!”