Fermat's Last Theorem

Published in the US as Fermat’s Enigma. This is the book I would give to someone who thinks mathematics has no stories. It is a 350-year detective story with a single, very simple question at its centre.

The question

Pythagoras gives us whole numbers with x² + y² = z², like 3, 4, 5. Fermat claimed, in the 1630s, that there are no whole-number solutions to xⁿ + yⁿ = zⁿ for any power n greater than 2. He scribbled in the margin of his copy of Diophantus that he had a marvellous proof which the margin was too small to contain. Nobody ever found it.

The relay race

What I liked most is how Singh turns the history into a relay. Fermat himself left a proof for the fourth powers, using his method of infinite descent: if a solution existed, you could always build a smaller one, which is impossible for positive whole numbers. Euler handled cubes. Sophie Germain, working under a male pseudonym because women were shut out, proved results that covered whole families of exponents. Lamé announced a full proof in Paris, only for Kummer to point out that it quietly assumed a kind of unique factorisation that fails for the numbers he was using.

Each of these is a near-miss, and each leaves behind new mathematics. That, more than the theorem itself, is the real lesson of the book.

The twist

The final solution comes from somewhere completely different. In the 1950s two Japanese mathematicians, Taniyama and Shimura, suggested that two unrelated-looking kinds of objects, elliptic curves and modular forms, are secretly the same thing. Decades later Gerhard Frey and Ken Ribet showed that a counterexample to Fermat would produce an elliptic curve that could not be modular. So proving the Taniyama–Shimura conjecture, at least for the right class of curves, would prove Fermat.

Andrew Wiles then spent about seven years on it, largely in secret. He announced a proof in Cambridge in 1993; a gap was found during review; with Richard Taylor’s help he repaired it, and the proof was published in 1995. Singh’s account of the year with the gap is the emotional peak of the book, and it is genuinely tense even though you know the ending.

Limits

You will not understand the proof from this book; nobody could, at this length. Singh is open about that and explains the shape of it instead. A few of the historical anecdotes feel polished for the telling.

Verdict

The best popular maths book I have read for sheer narrative. I finished it wanting to understand modular forms, which is probably the highest compliment a book like this can get.