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Portrait of Fermat's Last Theorem

Science · Mathematics

Fermat's Last Theorem

c. 1637 – 1995

Mathematics — number theory

The assertion that x^n + y^n = z^n has no solutions in positive whole numbers when n is greater than 2. Pierre de Fermat scribbled it in the margin of his copy of Diophantus's Arithmetica around 1637, adding that he had a truly marvellous proof which the margin was too narrow to contain. It resisted every attack for 358 years until Andrew Wiles, working largely in secret in his Princeton attic, proved the modularity theorem for semistable elliptic curves; his 1993 announcement contained a gap, repaired with Richard Taylor, and the complete proof appeared in the Annals of Mathematics in 1995.

How to spot them

  • ▸No whole-number solutions to x^n + y^n = z^n for n greater than 2
  • ▸The margin of a copy of Diophantus's Arithmetica, too narrow for the proof
  • ▸Proved by Andrew Wiles in 1993–95 after seven years of secret work
  • ▸The route went through elliptic curves, modular forms and the Taniyama–Shimura conjecture
  • ▸Wiles was too old for the Fields Medal and received a silver plaque instead

Key facts

A marginal note in a copy of Diophantus's Arithmetica, claiming a proof too long to fit (c. 1637)

Pierre de Fermat's annotation, found after his death by his son Samuel

A 1993 Cambridge lecture series ending 'I think I'll stop here' (1993)

Andrew Wiles at the Isaac Newton Institute, announcing the proof

The conjecture linking elliptic curves to modular forms, whose proof settled the question (1955)

The Taniyama–Shimura(–Weil) conjecture, now the modularity theorem

Ribet's 1986 theorem connecting a hypothetical counterexample to a curve that cannot exist (1986)

The epsilon conjecture, following Gerhard Frey's insight of 1984

A prize established by a German industrialist in 1908 and finally paid out in 1997 (1908)

The Wolfskehl Prize, awarded to Wiles by the Göttingen Academy

A 1994 collaboration with Richard Taylor that repaired a gap in the Euler system argument (1994)

The two papers appeared together in the Annals of Mathematics in May 1995

A 1996 BBC Horizon documentary and Simon Singh's bestseller about a 358-year problem (1996)

Horizon's 'Fermat's Last Theorem' won a BAFTA; Singh's book followed in 1997

A joke in The Simpsons: 1782^12 + 1841^12 = 1922^12, a near miss that fits a calculator display

Written by David X. Cohen; the equality fails, but not on a ten-digit display

Key ideas

  • The case n = 2 has infinitely many solutions — the Pythagorean triples. Everything above 2 has none, which is why the theorem is so tantalising.
  • The proof does not work with the equation directly. Frey showed a counterexample would give a strange elliptic curve; Ribet showed such a curve could not be modular; Wiles proved that semistable elliptic curves are all modular. The counterexample therefore cannot exist.

Quick-fire facts

  • •Pierre de Fermat (1607–1665) was a Toulouse lawyer and magistrate who did mathematics as a pastime
  • •The marginal note was published by his son Samuel in 1670, in an edition of Diophantus's Arithmetica
  • •Fermat did leave a proof for the case n = 4, by his method of infinite descent
  • •Euler dealt with n = 3, Sophie Germain made wide-ranging progress, and Kummer's theory of ideal numbers settled all regular primes
  • •Andrew Wiles announced the proof at the Isaac Newton Institute, Cambridge, on 23 June 1993
  • •A gap found in the autumn of 1993 was closed with Richard Taylor; the two papers appeared in the Annals of Mathematics in May 1995
  • •Wiles was 41 when the proof was completed, over the Fields Medal's age limit of 40; the IMU gave him a special silver plaque in 1998
  • •He received the Wolfskehl Prize in 1997, the Abel Prize in 2016, and was appointed a Knight Commander of the Order of the British Empire in 2000
  • •In September 2026 Anthropic announced that its Claude model had produced the first complete machine-checked proof in the Lean 4 proof assistant, reviewed by Kevin Buzzard, whose EPSRC-funded 'Formalising Fermat' project at Imperial College London had begun the formalisation in 2024
  • •Simon Singh's 1997 book and the 1996 BBC Horizon film made the story famous to non-mathematicians
  • •The related abc conjecture, if proved, would give another route to the result

Drill it — 9 cards

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Sources