Fermat's Last Theorem for 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, in the form that $x^4 - y^4 = z^2$ has no solution in positive integers. The proof uses infinite descent and the classification of Pythagorean triples. Since $4$ divides every power of $2$ above $2$, the case $n = 4$ settles Fermat's Last Theorem 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.