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. 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 and n = 3 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.