Fermat's Enigma, in detail
In 1637, Pierre de Fermat scrawled a note in the margin of a mathematics book claiming to have found a proof that no three positive integers can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2 — but that the margin was too narrow to contain it. That marginal note became the most famous unsolved problem in the history of mathematics, and it remained unsolved for 358 years. Simon Singh tells the story of how it was finally resolved in 1994, when Princeton mathematician Andrew Wiles completed a seven-year secret project to produce a proof spanning more than 100 pages.
The book works on two levels simultaneously. On one level it is intellectual history: Singh traces three and a half centuries of mathematicians who devoted their careers to the problem, from Euler and Gauss to Kummer, Mordell, and finally Wiles. He explains the mathematics with rare clarity — elliptic curves, modular forms, the Taniyama-Shimura conjecture, and the Galois representations that Wiles used to bridge them — in ways that a reader with no formal mathematics training can follow. On another level it is a human drama: Wiles spent seven years working in secret, avoiding conferences and declining to mention his work for fear of competition, and his announcement in June 1993 — followed by the discovery of a flaw, the 14 months of despair, and finally the fix — is one of the great scientific narratives of the 20th century.
Singh is a physicist and documentarian who made a BBC documentary about Wiles before writing this book, and the access he had to Wiles gives the account an intimacy unusual for a mathematics book. Wiles describing the moment he found the fix — "it was so indescribably beautiful" — carries weight that a more distant narrative wouldn't achieve.
The book does more than tell one story. It uses Fermat's Last Theorem as a lens through which to explain number theory more broadly, making the case that the pursuit of pure mathematical knowledge has intrinsic value independent of any application. The proof of Fermat's Last Theorem has no known practical use. Singh argues implicitly, and Wiles argues explicitly, that this is irrelevant. The problem was beautiful and it mattered.
The big ideas
- 1.
Fermat's Last Theorem states that a^n + b^n = c^n has no integer solutions when n is greater than 2. Fermat claimed a proof; no proof was found for 358 years.
- 2.
Andrew Wiles's proof works by establishing the Taniyama-Shimura conjecture for semistable elliptic curves, which had been shown to imply Fermat's Last Theorem by Ribet in 1986.
- 3.
Elliptic curves and modular forms are two apparently unrelated areas of mathematics that Taniyama and Shimura conjectured were secretly the same objects — a bridge between two mathematical worlds.
