1993: Fermat's theorem solved

   A shy and secretive Princeton University mathematics professor in 1993 unraveled a mystery that had frustrated and intrigued mathematicians for 350 years.

    Andrew Wiles, fascinated by math problems since age 10, figured out the last theorem of 17th century mathematician Pierre De Fermat, achieving what the most obsessed numbers crunchers of three centuries could not.

    The Scottish-born Wiles, in a rare interview, said the draw to solve the theorem, which stemmed from Fermat's studies of the ancient Greek text "Arithmetic," was so strong because the theorem was so simple-sounding.

    It says that while the square of a whole number can be broken into two other squares of whole numbers, the same cannot be done with cubes or higher powers.

    The theorem is based on the ancient equation developed by sixth century mathematician Pythagoreas, "X squared plus Y squared equals Z squared." The equation guided Pythagoreas' famous theory for calculating the hypotenuse of a triangle.

    Although Fermat himself claimed to have already proved the theorem, his notes were lost, and mathematicians, none of whom were able to solve it until Wiles, had often doubted the existence of a formal proof.

    Fermat's assertion that no powers higher than two fit the equation seemed simple to 10-year-old Wiles when he first stumbled upon the problem in a math book at his local public library more than 30 years ago.

    "No one knew if there was a proof. And people ever since had looked for the proof. And there was this problem that I, a 10-year-old, could understand, that none of the great mathematicians in the past had been able to resolve. And from that moment on, of course, I just tried to solve it myself. It was such a challenge, such a beautiful problem," Wiles said in a 1998 NOVA documentary on his accomplishments.

    Wiles, still a professor at Princeton University, rarely grants interviews, and could not be reached for comment on this article.

    Fermat, also known as a lawyer and a diplomat, made some of the greatest mathematical breakthroughs in history, most of which were scribbled down in the margins of his copy of "Arithmetica." All of his original notes were lost, but can be read in a book published by his son years after his death.

    Some of the missing notes may have contained proof of the theorem that had perplexed mathematicians for centuries, historians speculate.

    Over the past centuries, mathematicians one by one have proved the theorems contained in Fermat's notes. The theorem solved by Wiles was the last one and so referred to as "Fermat's Last Theorem."

    Any mathematician will attest that, no matter how many numbers are plugged into a formula, there will be an infinite amount of numbers left to test. A proof of Fermat's last theorem would at best provide a logical demonstration of why no other numbers fit into the equation without actually checking every single number.

    After centuries of failing to prove the last theorem, mathematicians in the 1970s began to abandon Fermat's work. Even Wiles, then a research student at Cambridge University, took leave from his work on the theorem, saying that his research was not generating other mathematical questions, a sign of a "bad math problem."

    "...and at that point I really put aside Fermat. It's not that I forgot about it; it was always there. I always remembered it, but I realized the only techniques we had to tackle it had been around for 130 years, and it didn't seem they were really getting to the root of the problem," Wiles said in the documentary.

    He began studying the "in" math of the early 1980s, elliptic curves, under the tutelage of Cambridge professor, John Coates.

    Elliptic curves are cubic curves whose solutions have shapes like donuts. Every point on the "donut" is the solution to an equation. The conjecture (unproved idea) of some mathematicians working on elliptic curves on the other side of the world would become the key to Wiles' history-making proof.

    When Wiles learned that two mathematicians in Tokyo, Goro Shimura and Utaka Taniyama, had developed an idea about elliptic curves being modular forms that could prove or disprove Fermat's Last Theorem, he was "electrified."

    He saw a connection between the conjecture and Fermat's Last Theorem that eluded other mathematicians.

    "I knew at that moment the course of my life was changing, because this meant that to prove Fermat's Last Theorem, I just had to prove the Taniyama-Shamura Conjecture," Wiles said in the documentary. "From that moment on, that was what I was working on. I just knew I would go home and work on the Taniyama-Shamura Conjecture."

    Most mathematicians, including Coates, who was somewhat of a mentor to Wiles, thought it impossible to prove the conjecture, let alone the illusive theorem of Fermat. None saw the connection between the two theorems — "Every elliptic curve is modular" and "No number fits this equation."

    Coates said in the documentary, "I must confess, I didn't think that the Shamura-Taniyama conjecture was accessible to proof at present. I thought I wouldn't see a proof in my lifetime. Andrew Wiles is probably one of the few people on Earth who had the audacity to dream that you could actually go and prove this conjecture."

    Wiles abandoned all other research, cut himself off from the world, and for seven years worked on proving the conjecture and, subsequently, Fermat's Last Theorem.

    While most researchers consult colleagues and keep their field abreast of progress in research, Wiles worked in isolation and secrecy, surprising others in his field when he revealed the fruits of his intellectual labor years later.

    A mathematician working for such an extended period of time on such a large project without discussing it with colleagues was unprecedented, other mathematicians said when he finally revealed the proof in September 1993.

    "In this case, certainly in the first few years, I had no fear of competition. I simply didn't think I or anyone else had any real idea how to do it. But I realized after a while that talking to people casually about Fermat was impossible because it generates too much interest, and you can't really focus yourself for years unless you have this undivided concentration which too many spectators would have destroyed," Wiles said.

    So he worked furiously on the proof, revealing his close proximity to the solution to only two colleagues at the cusp of completing the proof. When he was ready, in the fall of 1993, he called a conference at Cambridge with some of the greatest math minds of this era, and he revealed that he had proved the theorem. Although he didn't share the exact proof, reporters and curious mathematicians from around the world began calling him the very next day.

    He distributed the 200-page proof for review by other leading mathematicians and, after countless, easily-answered queries over e-mails and fax, one turned out to be a valid glitch in the proof.

    By the time the mistake was discovered, Wiles had already become an overnight celebrity — the man who had solved the most complex mathematical mystery in history. He was uncomfortable with the fame considering he did not technically prove the theorem with a large chunk of information still missing from the logical sequence.

    He finally fixed the proof a year later with the help of a former Princeton student, Richard Taylor.

    For his research he received an unprecedented honor from the International Mathematical Union during the International Congress of Mathematicians in Berlin last year. He was awarded a silver plaque in lieu of the Fields Medal, the highest honor in mathematics. The congress gave him the plaque because the medal is only awarded to mathematicians under 40, and Wiles was just over 40 when he solved the theorem.

    "There's no other problem that will mean the same to me. I had this rare privilege of being able to pursue in my adult life what had been my childhood dream. I know it's a rare privilege, but if one can do this, someone can actually tackle something in adult life that means so much to you, it's more rewarding than anything I could imagine," Wiles said.