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Fermat's Last Theorem states that the equation,

$a^n + b^n = c^n$

Has no positive integer solutions for all $n>2$. Fermat first proposed his theorem in 1637, where he noted in his copy of Arithmetica by Diophantus, that,

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain

Fermat never proved his theorem. The complexity of the solution leads most of the mathematics community to assume that Fermat was mistaken, and did not in fact have a proof. This would come as no surprise, as mathematicians often worked in isolation scarcely consulting others, leaving them prone to errors. Fermat's Last Theorem is named as such as it was the last of Fermat's theorems that was left unproved. Many individuals were quoted as saying that, perhaps, Fermat's Last Theorem was outside the realms of contemporary mathematics. While proofs for specific cases of Fermat's Last Theorem, namely $n=3$ and $n=4$, a proof that generalised for all $n$ was not obtained until 1994.

## Proof

Andrew Wiles was fascinated with Fermat's Last Theorem for a young age, but soon gave up his efforts at proving it when he discovered that his grasp of mathematics was insufficient. In his adult years, he stumbled across the Taniyama-Shimura conjecture, which was a conjecture involving elliptic curves. A proof of the Taniyama–Shimura conjecture, known as the Modularity Theorem once it was proved, would be deemed sufficient, in combination with other theorems, to prove Fermat's Last Theorem. Wiles first submitted a proof in 1993. When the proof was subjected to rigorous peer review, a fatal error was discovered which nullified the proof. Wiles, while slightly disappointed, was unphased, and addressed the issues in collaboration with Richard Taylor.

Fermat's Last Theorem was successfully proved in 1995.