# Fermat's Last Theorem

**Type:** Theorem  
**Domain:** Named Results · Number Theory  
**Conjectured:** 1637, Pierre de Fermat  
**Proved:** 1994, Andrew Wiles (published 1995)  
**Codex URL:** /mathematics/named-results/fermats-last-theorem/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Statement

No positive integers a, b, c satisfy aⁿ + bⁿ = cⁿ for any integer n > 2.

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## Timeline

| Year | Event |
|------|-------|
| 1637 | Fermat states conjecture in margin note |
| 1770 | Euler proves n=3 |
| 1825 | Legendre and Dirichlet prove n=5 |
| 1839 | Lamé proves n=7 |
| 1850s | Kummer proves regular primes |
| 1994 | Wiles completes proof (Taylor fixes 1993 gap) |

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## The proof strategy

Frey (1984): a Fermat solution would produce a non-modular elliptic curve (the "Frey curve"). Ribet (1986) proved this rigorously. Wiles proved the Taniyama–Shimura conjecture for semistable elliptic curves (1993–94, with Taylor), which by Ribet's result implies Fermat's Last Theorem. Full Taniyama-Shimura-Weil (Modularity Theorem) completed 2001 (Breuil, Conrad, Diamond, Taylor).

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## Sources

### Tier 1
- Wiles, A. (1995). Modular elliptic curves and Fermat's Last Theorem. *Annals of Mathematics*, 141(3), 443–551.
- Taylor, R. and Wiles, A. (1995). Ring-theoretic properties of certain Hecke algebras. *Annals of Mathematics*, 141(3), 553–572.

### Tier 2
- Cornell, G., Silverman, J.H. and Stevens, G. (eds.) (1997). *Modular Forms and Fermat's Last Theorem*. Springer.

### Tier 3
- Singh, S. (1997). *Fermat's Enigma*. Walker & Company.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*