Fermat’s Last Theorem
Author: Simon Singh | Published: 1997
Summary
Fermat’s Last Theorem tells the story of the most famous problem in mathematics—Pierre de Fermat’s 1637 marginal note claiming that he had a proof that no positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for n > 2 (generalizing the Pythagorean theorem)—and the 350-year search for that proof, culminating in Andrew Wiles’s 1995 proof using elliptic curves and modular forms. Singh, a science journalist and documentary maker, structures the book as a thriller: the historical chapters tracing the problem from Fermat through Euler, Sophie Germain, Kummer, and Wolfskehl are interspersed with the account of Andrew Wiles’s secret seven-year effort at Princeton and the dramatic error that surfaced after his initial announcement, the year spent by Wiles and his former student Richard Taylor fixing the error, and the final triumph.
The book makes a genuine attempt to explain the mathematics: not just the history of who worked on the problem and what techniques they developed, but why the problem was so hard, what modular forms and elliptic curves are and why the Taniyama-Shimura conjecture (the key bridge Wiles exploited) connects them to Fermat’s equation. Singh writes for readers with no mathematical background but with genuine curiosity, and the explanations—using vivid analogies and careful building of concepts—succeed better than most popular mathematics books. The emotional arc of the story—Wiles’s childhood dream, his secret years of work, the devastating error, the year of repair, the final proof—is genuinely gripping.
Fermat’s Last Theorem was the book that demonstrated to a generation of readers and publishers that mathematics could be made the subject of compelling popular narrative. It set a standard for “popular mathematics” as a genre—a genre that Singh, Marcus du Sautoy, Ian Stewart, and others have since developed—and showed that the human drama of mathematical research is as gripping as any other kind.
Critical Takeaways
- The Wiles proof: The proof Wiles ultimately produced was not the elementary proof Fermat claimed (if such a proof exists, it remains undiscovered); it uses 20th-century mathematics—elliptic curves, modular forms, Galois representations—that Fermat could not have known.
- Taniyama-Shimura: The key insight was that Fermat’s Last Theorem follows from the Taniyama-Shimura-Weil conjecture (now Modularity Theorem) relating elliptic curves to modular forms; the proof of Fermat is a corollary. This connection is one of the most unexpected in number theory.
- Sophie Germain and the history of women in mathematics: The chapter on Sophie Germain—who worked on Fermat under a male pseudonym and made real mathematical contributions—is one of the most affecting in the book and one of the clearest accounts of structural exclusion in the history of science.
- Genre-defining book: Fermat’s Last Theorem is credited with establishing popular mathematics as a viable publishing category; it demonstrated that mathematical narrative could reach mass audiences without compromise of serious content.
- The year-long error: The period between Wiles’s initial announcement and the corrected proof—during which a serious error was found—is the most humanly compelling part of the story and the most honest about what mathematical research actually looks like.
My Takeaways
- The description of Wiles’s reaction when he finally understood the correction to his proof—he sat staring at it for 20 minutes because it was so beautiful—is my favorite account of mathematical discovery in any popular source.
- The story of Sophie Germain—contributing real mathematics while concealing her gender, being acknowledged by Gauss only after her identity was revealed—made the history of structural exclusion in mathematics concrete and specific.
- The 350-year accumulation of failed approaches—each technique developed to attack Fermat generating new mathematics as a byproduct—is a model for how mathematical progress often works: the unsolvable problem generates solvable sub-problems that collectively build the tools for the solution.
- The Taniyama-Shimura bridge—a connection between two apparently unrelated areas of mathematics—is the aesthetic heart of the proof and demonstrates something about mathematical reality: deep connections exist between domains that appear separate.