Concept
No integer solutions for x^n + y^n = z^n when n>2 — Fermat's 1637 margin note, finally proved by Wiles in 1994.
Understand it in one breath
"x² + y² = z² has infinitely many integer solutions, but for n ≥ 3 there are none." Fermat's marginal note became public in a posthumous 1670 edition. Wiles worked privately from 1986, announced a result in 1993, and repaired a gap with Richard Taylor in 1994. The two papers appeared in the 1995 Annals of Mathematics.
At a glance
Year | Person | Advance |
|---|---|---|
1637 | Fermat | Conjectured in a book margin (a copy of Diophantus) |
1770 | Euler | Proof for n = 3 (though a gap was later found) |
1825 | Dirichlet and Legendre | n = 5 |
1839 | Lamé | n = 7 (using Germain’s theorem) |
1908 | Wolfskehl Prize | 100,000-mark prize — inspired countless false proofs |
1955 | Taniyama–Shimura conjecture | Conjectured that every elliptic curve is modular |
1982 | Frey | Proposed that an FLT counterexample ⇒ a non-modular elliptic curve |
1986 | Ribet | Proved the epsilon conjecture → completed the bridge between FLT and Taniyama–Shimura |
1993 announcement / 1994 repair / 1995 publication | Andrew Wiles (+ Richard Taylor) | Two papers completed the argument after years of concentrated private work |
The simplest-looking statement took 358 years to solve — and the solution came from entirely different fields: elliptic curves and modular forms.
Key formula
Worked examples
- 1
Q.n=2: 3² + 4² = ?
- 2
Q.n=3: are there positive integer solutions to x³ + y³ = z³?
Key moments
A note in the margin
Fermat wrote in the margin of his copy of Diophantus: “I have discovered a truly marvelous proof, which this margin is too narrow to contain.”
Euler proves the case n=3
Euler used infinite descent to establish the n=3 case, but even he could not reach a general proof.
The 100,000-mark Wolfskehl Prize
Paul Wolfskehl endowed a 100,000-mark prize for a proof, drawing generations of professional and amateur attempts.
Wiles — the end of 357 years
After seven years of secret work and another year repairing a critical gap, Andrew Wiles completed the proof — 357 years after Fermat’s marginal note.
Modern applications
A driving force behind algebraic geometry, Galois representations, modular forms, and the broader toolkit of modern number theory. The mathematics built for the proof is an even larger legacy than the theorem itself.
Beyond MathVoyage
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