Pythagorean triples give infinitely many whole-number solutions to a^2 + b^2 = c^2. Fermat asked the obvious next question — what about cubes, fourth powers, higher? — and scribbled an answer in a margin that would torment mathematics for 358 years.

The statement

The leap from n = 2 (infinitely many solutions) to n = 3 (none at all) is staggering. In 1637 Fermat wrote that he had "a truly marvellous proof, which this margin is too narrow to contain." No such proof was ever found among his papers — and almost certainly none existed.

Centuries of partial progress

The theorem was chipped away exponent by exponent. Fermat himself settled n = 4 by infinite descent; Euler did n = 3. Attempts to factor a^n + b^n using algebraic integers ran aground on a shocking discovery — unique factorisation can fail in those larger rings. Kummer's work to repair it, inventing "ideal numbers", founded algebraic number theory — so the failed assaults built an entire field.

The resolution

The proof finally arrived in 1994, from Andrew Wiles, by a route Fermat could never have imagined: it goes through elliptic curves and modular forms, proving a deep case of the Taniyama–Shimura conjecture. The argument runs to over a hundred pages of modern machinery. Fermat's Last Theorem is the perfect emblem of number theory — a question a child can understand, whose answer required inventing centuries of new mathematics.