In 1993 Princeton University professor Andrew J. Wiles (born 1953) announced that he had solved one of the most legendary challenges in mathematics. Fermat's Last Theorem was an elegantly simple problem in need of a proof, and it had confounded mathematicians both professional and amateur for some 350 years. Wiles's successful cracking of the necessary code caused a stir in the math community and even landed him on the front page of the New York Times for solving what Science writer Barry Cipra asserted was "one of the unconquered peaks of mathematics."

# Andrew J. Wiles

Andrew J. Wiles was born April 11, 1953, in Cambridge, England, where his father was professor of theology at the famed medieval university there. In the Cambridge library the ten-year-old Wiles first came across Fermat's Last Theorem, and it intrigued him. He worked on it in his teens before realizing it was far more complex a challenge than he had originally assumed. After earning an undergraduate degree from Oxford University in 1974, Wiles went on to earn graduate degrees in math from Clare College, Cambridge, and specialized in elliptical curves, a relatively new field of higher math. In 1982, two years after he earned his Ph.D., he began teaching at Princeton University in New Jersey.

## Boyhood Fascination Rekindled

Although Wiles was still intrigued by Fermat's Last Theorem, he knew it would be foolish to devote time and energy to it. Many, many minds before him had failed. The most successful had been Ernst Eduard Kummer, whose development of algebraic number theory led to some notable advances in solving the puzzle in the 1870s. As Wiles told *Sciences* writer Peter G. Brown, he had given up on it himself by graduate school. "I think I also must have realized that it was not a good idea, for the reason that not much new theory had been developed to deal with the problem since Kummer… . You don't want to waste your whole life
as a mathematician trying to find some bizarre, ephemeral proof."

## The Missing Proof

Fermat's Last Theorem dates back to 1637. "What makes the theorem so tantalizing is that for all its fiendish difficulty to prove, it is almost absurdly simple to state," noted *Time* contributor Michael D. Lemonick. Pierre Fermat was a brilliant self-taught mathematician and lawyer living in France. Before his 1665 death he made several important advances in probability theory and analytic geometry theory. The theorem that bears his name had its origins in a note he scribbled, as was his habit, in the margin of a copy of Diophantus's *Arithmetic,* a treatise dating from third-century Greece. In short, Fermat asserted that "a to the nth
power + b to the nth power = c to the nth power" can never be true when the exponent "n" is greater than two. In a way, it is similar to the well-known Pythagorean theorem, which holds that the square of the longest side of a right triangle is equal to the sum of the squares of the other two sides. Fermat, explained *Guardian* writer Simon Singh, "created the equation [x to the nth power + y to the nth power = z to the nth power], where "n" represents any number bigger than 2. Fermat came to the extraordinary conclusion that this new equation had no solutions— among the infinity of numbers, none existed that fitted his equation."

Fermat did not leave behind a "proof," or solution, proving his assertion, noting instead that he had found one that was very simple, but the margin of the book was too small for him to write it down. That boast incited a 350-year-quest to find it, which would end with Wiles's historic solution. As Wiles explained to Singh: "Pure mathematicians just love a challenge. They love unsolved problems. Most deceptive are the problems which look easy, and yet they turn out to be extremely intricate. Fermat is the most beautiful example of this. It just looked as though it ought to have a simple proof and, of course, it's very special because Fermat said that he had a proof."

## Lured the Learned, the Daft

Proving Fermat's theorem had confounded generations of Wiles's predecessors. In 1780 Leonhard Euler found that an exponent of three would not work, and others found that exponents of 5, 7, and 13 cannot be true either. The maddening problem so intrigued German industrialist Paul Wolfskehl that he offered a large cash prize to anyone who could solve it. His announcement in 1907 sent a flood of solutions to a special prize commission office established at the University of Göttingen, every single submission required being checked. There were a dwindling number of entries each decade, but a few still came in every month as the 100th-anniversary deadline to solve it—September 13, 2007—neared. In some cases the submissions were from qualified researchers, noted Singh, adding that, in other cases, "manuscripts bore clear signs of schizophrenia." Some respected names in the field began to theorize that the proof Fermat mentioned never existed, or that he recognized its serious flaws and destroyed it. Writing in *Sciences,* Brown described Fermat's Last Theorem as "a siren call for the unwary since the seventeenth century. Amateur and professional mathematicians alike have been lured into its quicksand, many to give up, after years of effort, in frustration and disgust."

With the dawn of the computer age, programmers ran calculations in an attempt to solve Fermat's Last Theorem up to the number 4,000,000, but little real advancement was made. In 1984 a panel of number theorists declared it would never be proved or disproved. Characteristically, the final solution was linked to Wiles's chosen discipline. "One newish branch of algebraic geometry deals with a group of shapes known as elliptical curves, most of which look like a wiggly hump with an egg on top," explained an *Economist* contributor. "It is by manipulating such curves that mathematicians now find they can infer various things about statements such as Fermat's last theorem."

## Solution Buried in Another Riddle

In the mid-1950s Yutaka Taniyama, a Japanese mathematician, asserted that an elliptic curve has modular form; this idea was picked up in 1971 by another Japanese mathematician, Goro Shimura and came to be called the Taniyama-Shimura conjecture. A conjecture is an intriguing but unproven theory. Little else came out of this idea until the early 1980s, when an academic in Saarbrucken, Germany, named Gerhard Frey issued a paper asserting the key to proving Fermat's Last Theorem was in the Taniyama-Shimura conjecture. Frey stated that an elliptical curve could represent all the solutions to Fermat's equation; in other words, if Fermat's Last Theorem were false, there would be elliptic curves that violated the Taniyama-Shimura conjecture. A University of California at Berkeley mathematician, Kenneth Ribet, agreed with this idea. In 1986 "Ribet showed that if Fermat's theorem is wrong, then some elliptic curves should exist that could not be constructed according to Taniyama's conjecture," explained *Discover* writer Tim Folger.

Upon learning of Ribet's announcement, Wiles set out to prove that such curves exist. He went back to work on solving Fermat's Last Theorem that same day, telling no one save for his wife and one trusted colleague. He spent seven years working on it in the attic office of his home in Princeton, leaving only to spend time with his family, which includes two daughters, and to teach classes. He abandoned all of his other work to concentrate on it, and was rarely seen at professional conferences. In an interview with *Science* contributor Cipra, he likened the process to "entering a darkened mansion. You enter a room, and you stumble months, even years, bumping into the furniture. Slowly you learn where all the pieces of furniture are, and you're looking for the light switch. You turn it on, and the whole room is illuminated. Then you go on to the next room and repeat the process."

## Historic Announcement Made

In 1991, after five years of concentrating how to find the solution, Wiles made the breakthrough that set him on the right path. He reduced the theorem to a calculation that had been used unsuccessfully by others and, as he told Cipra, became convinced that "the proof was just around the corner … but the comer was a bit longer than I anticipated." In May of 1993 he felt that he had the proof nearly complete, save for one part. At that point he came across a paper from Harvard mathematician Barry Mazur that described one type of mathematical construction, and Wiles used it to get through the final roadblock. This last step took him just six weeks.

Wiles decided to announce his finding at a series of lectures held at Cambridge University. Their title, "Modular Forms, Elliptic Curves, and Galois Representations," did not give any hint of the historic revelation he was about to make, but Wiles had been out of sight for so long that rumors abounded days before his lecture series. On the first day Wiles recounted the first five years of his work on the Taniyama-Shimura conjecture. The second day he presented his findings from the 1991 to 1993 period. On the final day he summed up, with copious blackboard notations, his last six weeks of work. On that day, June 23, 1993, he concluded by telling the assembled mathematicians that he had proven the Taniyama-Shimura conjecture, and noted, in a casual aside, that it meant Fermat's Last Theorem was also proven to be true. The audience burst into tremendous applause, and news quickly circulated throughout the scientific community.

Yet Wiles refused to immediately release his 200-page proof to his international colleagues for verification, and rumors arose that there was a flaw near the end of it. In December of 1993 he announced that there was indeed a problem, but that he planned to solve it himself. "The problem occurred in Wiles's construction of a mathematical object known as an Euler system, which is a relatively new and largely unexplored idea," explained Cipra. "Wiles's Euler system was intended to prove a sizable chunk of the Taniyama-Shimura conjecture … but the system he had come up with turned out not to work in quite the right way."

## After 90 Years Prize Awarded

On September 19, 1994, Wiles had a eureka moment and closed the theoretical gap. "It was so simple and elegant that at first it seemed too good to be true," he recalled to *Science* interviewer Cipra. A month later he finished and announced his corrections; the findings were published in the May 1995 issue of *Annals of Mathematics.* As befitting the historic nature of the proof, an entire issue was devoted to it, one part containing Wiles's paper for the original 1993 theory and a shorter one written by his former student, Richard Taylor, explaining how the flaw was overcome. As Brown remarked in *Sciences,* "in Wiles's proof Fermat's last theorem … tumbles out of a massive and intricate mathematical machine as a mere corollary, the consequence of a result that lies at the confluence of virtually every major stream of modern mathematics."

It took another two years to firmly double-check Wiles's proof of Fermat's Last Theorem—in part because there is a limited number of mathematicians who could grasp its complexities—before the Wolfskehl committee finally granted him the long-awaited prize. He received a number of other honors in his field and returned to Princeton where he continued teaching courses in number theory.

## Books

*Math and Mathematicians: The History of Math Discoveries around the World,* U*X*L, 1999.

*Notable Mathematicians,* Gale, 1998.

*Notable Scientists: From 1900 to the Present,* Gale, 2001.

Singh, Simon, *Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem,* Walker, 1997.

## Periodicals

*Discover,* January 1994.

*Economist,* July 3, 1993.

*Guardian* (London, England), June 26, 1997; July 22, 1999.

*Science,* December 24, 1993; November 4, 1994; May 26, 1995.

*Science News,* November 5, 1994.

*Sciences,* September-October 1993.

*Time,* July 5, 1993.