Perhaps the most important encounter with mathematics in my schooldays was when I found a copy of E.T. Bell’s ‘Fermat’s Last Theorem’ in my local library. As a ten year old I was immediately captivated by this three hundred year old problem. The question was clearly stated on the cover: can you prove that there are no solutions in rational numbers, none of which are zero, to the equation
xn + yn = zn
when n is an integer greater than or equal to three? Fermat had claimed in the margin of his copy of a book of the classical Greek mathematician Diophantus that he ‘truly had a wonderful proof but this margin is to small to contain it’.
I passed many hours of my childhood in trying to solve this problem. Neither of my parents was a mathematician but they were both sympathetic to the field. My mother had studied mathematics and physics and my father was a theologian but he always enjoyed puzzles and had been a codebreaker during the war. After undergraduate studies at Oxford I went to Cambridge as a graduate student in number theory, the branch of mathematics that tries to solve problems like the one of Fermat.
Together with my supervisor John Coates I worked on problems to do with elliptic curves. These problems go back a thousand years but the modern study of them began with Fermat. We were successful in making the first progress on a fundamental conjecture in the subject due to Birch and Swinnerton-Dyer.
After finishing my graduate studies in Cambridge I went on to my first position at Harvard, where after two years learning about modular forms and modular curves I began a very successful collaboration with Barry Mazur. This resulted in the resolution of the Iwasawa conjecture.
In 1982 after an interlude in Europe I moved to Princeton where I have remained, except for leaves abroad, ever since. For the first few years I pursued a generalization of what I had been doing at Harvard. However in 1986 after returning from a leave in Paris I heard the news that a new approach to Fermat’s Last Theorem had opened up, thanks to new ideas of Frey, Serre and Ribet. This transformed my working life. The methods that I had tried to use in childhood and during my student days were tired and when I had started graduate studies I had put them aside. Now I had a new opportunity to work on the problem this time using the theories of elliptic curves and modular curves. Moreover these were exactly the theories that I had been studying in my Cambridge and Harvard days. The challenge proved irresistible.
For the next seven years I worked on this approach to Fermat. The period was a private one for I soon found that it was inadvisable to discuss what I was working on. It was a period of intense work, searching for clues in what had been done, trying and retrying ideas until I could force them to take shape, a period of frustration too but punctuated by sudden thrilling insights that encouraged me to think I was on the right track. Then after five years I made a profound discovery. I could reduce the problem to a question that was precisely of the type I had studied in Harvard and during my first years at Princeton.
During the next two years I worked frantically to try to finish it and finally in May 1993 I believed I had done so. I presented the results of my work in June 1993 at a conference in Cambridge. At the end of the summer a problem was pointed out to me that led me to an error in one part of the proof, and I had to set about finding an alternative path for that section. It took me until September of 1994 to find the remedy, during which time I had the assistance first of a colleague Nick Katz (who had pointed out the first problem to me) and then of a former student of mine, Richard Taylor, with whom part of the final version was jointly written. The moment of illumination when I found the final key was one of unparalleled excitement and relief.
The year that I spent in correcting the argument was not an easy one. Happily during 1988 I had married my wife Nada and we had two daughters by the time of the Cambridge conference. Our third was born in May of 1994 in time for the final resolution. I cannot imagine that period without the support and demands of a family. It was hard to tear myself away from thinking about the problem every waking moment but fortunately my daughters managed to distract me just enough to keep some balance in my life.
The proof was published in May 1995 in the Annals of Mathematics, some 350 years after Fermat first wrote down the problem.
2 September 2005, Hong Kong
