---
tag: 0Y5X3SDBCZ
kind: statement
role: theorem
name: Fermat's Last Theorem
aliases: [flt]
msc: [11D41]
uses: [08B1RTW7VZ]
sources:
  - url: https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
    locator: Wiles's general proof
same_as:
  wikidata: Q132469
  wikipedia: Fermat's Last Theorem
---

## Statement

Let $n$ be an integer with $n > 2$.
Then there are no positive integers $a$, $b$ and $c$ with $a^n + b^n = c^n$.

## Remarks

For $n = 2$ there are infinitely many solutions, the [[Pythagorean triples|01G1WY1GVX]].
It is enough to prove the theorem for $n = 4$ and for each odd prime $n$, because each $n > 2$ has one of these as a divisor.
The cases [[n = 4|0E741HSMJ1]] and [[n = 3|07NKG38VEG]] have elementary proofs.

## History

Pierre de Fermat stated the theorem around 1637 and wrote that he had a proof.
Andrew Wiles gave the first proof in 1994, with a step completed jointly with Richard Taylor. The proof was published in 1995.
The proof uses the modularity of semistable elliptic curves.

## Explanations

### L1 human

The equation $a^2 + b^2 = c^2$ has many whole-number solutions, such as $3^2 + 4^2 = 5^2$.
For cubes, fourth powers and all higher powers, the theorem says that there are none.
The statement is simple, but the proof took more than 350 years.