Sir Andrew Wiles describes history behind famous equation

Math superstar packs ’em in

BY MIKE WILLIAMS
Rice News staff

With every seat in Grand Hall taken and hundreds standing in the aisles, mathematician Sir Andrew Wiles enjoyed the kind of reception normally afforded star athletes at the President’s Lecture Nov. 11.

But he is perhaps the Hank Aaron of intellectuals, having accomplished a mighty feat thought by so many for so long to be impossible.

Wiles visited Rice to tell ”The Story of an Equation,” specifically Fermat’s Last Theorem, the one mathematicians struggled with for centuries before the professor at Princeton and chair of its mathematics department finally proved it in 1994.

The talk was less about the British-born Wiles’ eight-year struggle to solve the puzzle that had entranced him since the age of 10, a story well-documented here, and more about where the theorem came from and why it became an obsession for mathematicians over the ages.

Wiles began with the very basics, describing the familiar Pythagorean theorem, x²+ y² = z², from which Fermat’s revelation springs. Wiles said the Babylonians had come up with the same concept 1,500 years earlier, writing the theorem and its solutions (in base 60) on clay tablets, though the concept of finding a proof for their solutions had yet to be invented.

That came later, as generations of mathematicians laid the basis for algebra and calculus, dueling each other to prove their supremacy and keeping their techniques secret to preserve their reputations.

”How does a duel work? Well, each one was supposed to set 30 problems for the other,” said Wiles of one such medieval math contest. ”They would be handed to lawyers — lawyers had some purpose — and then at the appropriate time they would both see each other’s solutions.” He described the prize in one such duel as ”30 lavish banquets,” but the real reward was an enhanced reputation.

Math had other perils, Wiles said, listing the nasty demises of a number of practitioners. Disease, murder and, in one case, Russian roulette all took their toll of famous mathematicians at young ages.

When Pierre de Fermat, a lawyer and judge as well as a mathematician, wrote in the margins of a math book in 1637 his thought that no solution could be found for the equation xⁿ+ yⁿ = zⁿ where n is greater than 2, he also noted he had ”a truly marvelous proof of this, which this margin is too narrow to contain.”

While the culture established by mathematicians through the ages didn’t necessarily require that he reveal his proof, Wiles doubted he had one at all. ”We don’t really believe Fermat could solve it,” said Wiles, noting the mathematical tools to pull it off just didn’t exist yet. In fact, had Wiles himself started trying to solve the problem 10 or 15 years earlier, he wouldn’t have had the tools, either. ”I came along at just the right time,” said the professor, whose own proof goes on for several hundred pages.

Wiles said Fermat is best remembered now as the founder of modern number theory. ”What’s truly remarkable is that he didn’t publish anything. What is known of him is from his correspondence and from his notes in the Greek book, ‘Arithmetica.'”

He noted Fermat made his name for his work on ”this incredibly strange phenomenon” called amicable numbers. ”You take the divisors of 220, they add up to 284. You take the divisors of 284, they add up to 220. Fermat was the first person in Europe to find the next pair of amicable numbers: 17,296 and 18,416. This is what he became famous for at the time,” said Wiles, pausing for effect. ”It is of absolutely no interest.”

Wiles said Fermat took delight in poking other less brilliant mathematicians. ”His correspondence consisted mainly of challenges to English mathematicians — and I was very happy to take him on.”

Having first discovered Fermat’s Last Theorem in a book about its history titled ”The Last Problem” by E.T. Bell (updated in 1998 to include news of the proof), the 13-year-old Wiles ”decided Fermat didn’t know any more sophisticated mathematics than I did. … I was going to try it.”

After college, he said, he made a conscious decision to put dreams of proving the theorem aside. ”I regarded it as an addiction I was going to stay away from. But then in 1985 something remarkable happened. A German mathematician named Gerhard Frey suggested a completely new way of attacking the problem.” That solution involved a new theory of elliptic curves, and it turned out that proving one theory contributed to proving Fermat as well. That drove Wiles back to his quest, which took seven years, with another year devoted to quashing a hitch in his solution.

When asked what prompted him to return to Fermat, Wiles said, ”What it takes is some kind of faith that, yes, this problem is solvable, and it’s solvable now. When Frey made his connection, I was convinced that not only was this problem doable, but it had to be done.”