Fermat's Last Theorem
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Proved by Andrew Wiles.
Exact statement
For any integer n ≥ 3, the equation xⁿ + yⁿ = zⁿ has no solutions in positive integers.
Status
Solved by Andrew Wiles (1994, published 1995), with a key step completed with Richard Taylor.
Posed: c. 1637, Pierre de Fermat.
In plain words
For n = 2 there are infinitely many solutions (3² + 4² = 5²). Fermat wrote in a margin that he had a proof for larger n that did not fit; it took 358 years. Wiles proved a case of the modularity theorem for elliptic curves, which by work of Frey, Serre and Ribet implies the theorem.
How it was solved
- c. 1640: Fermat proves n = 4 by infinite descent.
- 1770: Euler proves n = 3.
- 1986: Ribet proves that modularity of semistable elliptic curves implies Fermat's Last Theorem.
- 1995: Wiles, and Taylor–Wiles, published in Annals of Mathematics 141.
What our solver can and cannot do
The Fermat's Last Theorem is settled, so the solver shows the result and its history. It can check concrete examples, but the full proof lives in the references below. Open the solver.
Frequently asked questions
- What does the Fermat's Last Theorem say?
- For any integer n ≥ 3, the equation xⁿ + yⁿ = zⁿ has no solutions in positive integers.
- Has the Fermat's Last Theorem been solved?
- Solved by Andrew Wiles (1994, published 1995), with a key step completed with Richard Taylor.
- Can MathGPT solve the Fermat's Last Theorem?
- MathGPT shows the statement, who settled it and how, with references; the proof itself is in the cited papers.
References
- A. Wiles, 'Modular elliptic curves and Fermat's Last Theorem', Annals of Math. 141 (1995).
- R. Taylor and A. Wiles, Annals of Math. 141 (1995).
Related problems
- Riemann Hypothesis (unsolved)
- Birch and Swinnerton-Dyer conjecture (unsolved)
- Collatz conjecture (3n + 1 problem) (unsolved)
- Goldbach conjecture (unsolved)
- All famous problems