Fermat’s Last Theorem

1621 – 201638 items

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Pages 338–339 of Samuel de Fermat’s 1670 edition of Diophantus’s Arithmetica, with his father’s observation proving the case of fourth powers. Scan: Leonid. Public domain, via Wikimedia Commons.
Pages 338–339 of Samuel de Fermat’s 1670 edition of Diophantus’s Arithmetica, with his father’s observation proving the case of fourth powers. Scan: Leonid. Public domain, via Wikimedia Commons.

Three and a half centuries of one problem

Some squares add up to a square: 3² + 4² = 5², and there are infinitely many more like it. Around 1637 Pierre de Fermat, a judge in Toulouse, wrote in the margin of a book that the same never happens for higher powers: no whole numbers x, y and z make xⁿ + yⁿ = zⁿ when n is bigger than 2. He had, he said, a marvellous proof, but no room for it. Nobody found one until Andrew Wiles, in 1994, using mathematics Fermat could not have imagined.

How it is laid out

  • Fermat in Toulouse: the margin note, his proof for fourth powers, the 1670 edition that made the note public.

  • One exponent at a time: Euler, Sophie Germain, Dirichlet, Legendre and Lamé, then Lamé’s failed general proof and Kummer’s rescue.

  • The modular road: a conjecture from Tokyo about elliptic curves, Frey’s strange curve, Ribet’s bridge, and Wiles’s seven years in an attic in Princeton.

Filter by special cases for the exponents proved one by one, flawed proofs for the ones that fell apart, elliptic curves for the modern route, and prizes for the money and medals the problem attracted. Euler’s work in St Petersburg is dated in the Julian calendar, which Russia kept until 1918; everything else is Gregorian.

Fermat couldn’t possibly have had this proof. It’s a 20th century proof. — Andrew Wiles, NOVA, 1997

Items

  1. Bachet’s Latin Diophantus

  2. Fermat in Toulouse

    ≈ 33 years, 6 months · Toulouse

  3. The margin note

    Toulouse

  4. Fermat describes infinite descent

  5. Fermat dies at Castres

    Castres

  6. Samuel de Fermat prints the notes

  7. ≈ 82 years

  8. One exponent at a time

    ≈ 93 years

  9. Euler claims a proof for cubes

    Berlin

  10. Euler’s Algebra: cubes, with a gap

    St Petersburg

  11. Gauss declines to compete

    Göttingen

  12. Sophie Germain’s grand plan

    Paris

  13. Legendre prints Germain’s theorem

  14. Dirichlet and Legendre: fifth powers

    Institut de France, Paris

  15. Lamé proves the case of sevens

    Institut de France, Paris

  16. Lamé announces a general proof

    Institut de France, Paris

  17. Kummer’s ideal numbers and regular primes

    Breslau (now Wrocław)

  18. The Academy gives its medal to Kummer

    Institut de France, Paris

  19. ≈ 48 years, 8 months

  20. Paul Wolfskehl’s bequest

    Darmstadt

  21. The Wolfskehl Prize stands open

    ≈ 89 years · Göttingen

  22. The modular road

    ≈ 39 years, 8 months

  23. Taniyama’s problems at Tokyo–Nikkō

    Tokyo

  24. Taniyama dies

    Tokyo

  25. Andrew Wiles meets the problem at ten

    Cambridge

  26. Faltings: finitely many solutions

  27. Frey’s curve

    Mathematical Research Institute Oberwolfach

  28. Wiles’s seven secret years

    ≈ 6 years, 11 months · Princeton

  29. Ribet closes the hole

    Caffè Strada, College Avenue, Berkeley

  30. Wiles confides in Nick Katz

    Fine Hall, Princeton University

  31. “I think I’ll stop here”

    Isaac Newton Institute, Cambridge

  32. The gap

    ≈ 1 year, 1 month · Princeton

  33. Wiles admits the gap

  34. Richard Taylor joins the repair

    Fine Hall, Princeton University

  35. The fix

    Fine Hall, Princeton University

  36. Two papers in the Annals

  37. The Wolfskehl Prize is paid

    Göttingen

  38. A silver plaque in Berlin

    Berlin

  39. The whole conjecture falls

  40. The Abel Prize

    University Aula, Oslo