---
tag: 0E741HSMJ1
kind: statement
role: theorem
name: Fermat's Last Theorem for n = 4
msc: [11D25]
uses: [086BE0J3FZ, 0EFYCFHYFG]
sources:
  - url: https://en.wikipedia.org/wiki/Proof_of_Fermat%27s_Last_Theorem_for_specific_exponents
    locator: n = 4
---

## Statement

There are no positive integers $a$, $b$ and $c$ with $a^4 + b^4 = c^4$.

## Remarks

A stronger statement holds: $x^4 + y^4 = z^2$ has no solution in positive integers.
The theorem also follows from [[Fermat's right triangle theorem|086BE0J3FZ]], in the form that $x^4 - y^4 = z^2$ has no solution in positive integers.
The proof uses [[infinite descent|0EFYCFHYFG]] and the [[classification of Pythagorean triples|0KF3J2KAER]].
Since $4$ divides every power of $2$ above $2$, the case $n = 4$ settles [[Fermat's Last Theorem|0Y5X3SDBCZ]] for all such exponents.

## History

Fermat's proof of his right triangle theorem, by infinite descent, gives this case.

## Explanations

### L1 human

The equation $a^4 + b^4 = c^4$ has no whole-number solutions.
The proof assumes a solution and builds a smaller one from it, which leads to a contradiction.