American Scientist

The Fermat Diary. C. J. Mozzochi. xii + 196 pp. American Mathematical Society, 2000. $29.

When Pierre de Fermat, sometime in the 1630s, began to read Diophantus's Arithmetica, he probably had no idea that he was opening a chapter in the history of mathematics that would not be closed until more than 350 years later. Diophantus's book was very different from most Greek mathematics. Instead of geometry, it dealt with what we have since come to call algebra. Instead of theorems, it contained problems and solutions. Most of the problems asked for numbers (meaning either whole numbers or fractions, not arbitrary quantities) that were related in arithmetically interesting ways. As he read through the book and made notes in the margin, Fermat began to construct a whole new field of mathematics dedicated to numbers and their properties.

Fermat was an unusual mathematician. He made his living as a jurist, so he didn't need to play by the rules. He published essentially nothing, contenting himself with writing letters about his discoveries. Most of the time, he told people what he had been able to prove but did not share his proofs. It was only after his death that attempts were made to collect and publish his work. His son Samuel decided then to publish a new edition of Arithmetica incorporating his father's marginal notes into the text. It is because of this edition that we know about the most famous of those 30 or so marginal notes.

Next to a passage where Diophantus had solved the problem of expressing a square (such as 25) as the sum of two squares (such as 9 and 16), Fermat noted that on the other hand it was not possible to write a cube as the sum of two cubes, or a fourth power as the sum of two fourth powers, and so on for any higher power. In modern terms, the claim was that the equation xn + yn = zn had no solution in which x, y and z are positive whole numbers and n is at least three. "I have found a marvelous proof of this," he added, "but it will not fit in this margin."

So began 350 years of searching. In the 18th century, Leonhard Euler went through Fermat's work on numbers and established it as a coherent mathematical theory. He proved most of Fermat's assertions. He even found one (one!) assertion that was incorrect. He noted that Fermat himself had provided most of the proof that no fourth power can be written as a sum of two fourth powers, and he showed how to prove the assertion for cubes. But the general assertion remained out of reach. In time, it became known as "Fermat's Last Theorem"—not his last assertion, but the last one of his assertions still unproved.

Fast-forward to 1993. At a summer conference held at the University of Cambridge, Andrew Wiles announced that he had found the long-sought proof. Recent work had established a connection between Fermat's "Theorem" and another famous (although not nearly as old) conjecture, which Wiles had managed to prove.

Wiles's announcement was a huge event in the mathematical community. Just as huge, then, was the subsequent long silence. The announcement of a significant result is usually followed by the distribution of preprints, but in this case no manuscript appeared for a long time, and rumors of a problem started to fly. In December, Wiles acknowledged that there was a gap in his proof. Several months of speculation followed, during which the main ideas of Wiles's proof became known, but the details remained secret. Finally, in September of 1994, Wiles announced that the problem had been solved, and two papers, one by Wiles alone and one joint work with Richard Taylor, were circulated. Fermat's old claim had indeed finally been proved.

The story of Wiles's proof has already been the subject of several books, a television documentary and even a musical play. The book under review testifies to the mathematical community's continuing fascination with this episode. It is basically an outside observer's account of the goings-on: the lectures, the e-mail, the newspaper stories, the speculation. Mozzochi chronicles events from Wiles's 1993 announcement through 1995, when it became generally accepted that the proof was complete. He does not discuss the further work that led to the proof of the full Taniyama Conjecture. Throughout, Mozzochi touches lightly on the mathematical ideas, mostly to sort out who was responsible for what. He includes many photographs, a list of most of the people involved in one way or another, an excerpt from the introduction to Wiles's paper and a good bibliography.

Mozzochi provides us with a great deal of valuable primary data. The photographs, in particular, will be of great value to future historians. He doesn't, however, go much deeper than that. For background, for an analysis of the mathematical ideas or even for a discussion of the meaning of the various events discussed here, the reader will have to look elsewhere.—Fernando Q. Gouvêa, Mathematics, Colby College, Waterville, Maine