---
tag: 07NKG38VEG
kind: statement
role: theorem
name: Fermat's Last Theorem for n = 3
msc: [11D25]
uses: [0EFYCFHYFG]
sources:
  - url: https://en.wikipedia.org/wiki/Proof_of_Fermat%27s_Last_Theorem_for_specific_exponents
    locator: 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|0EFYCFHYFG]] 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|0Y5X3SDBCZ]].

## 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.