Fermat's Last Theorem for n = 3

Statement

There are no positive integers $a$, $b$ and $c$ with $a^3 + b^3 = c^3$.

Remarks

A sum of two positive cubes is never a cube. Standard proofs use infinite descent in the ring $\mathbb{Z}[\omega]$ of Eisenstein integers, where $\omega$ is a primitive cube root of unity. This case is part of Fermat's Last Theorem.

History

Fermat mentioned the case $n = 3$ in letters of 1636, 1640 and 1657. Euler published a proof in 1770. Many other proofs followed, for example by Legendre.

Explanations

L1 human

You can never write a cube as a sum of two positive cubes. For example, $6^3 = 216$, and $3^3 + 4^3 + 5^3 = 216$ needs three cubes, not two.