Sum of two Fourth Powers cannot be Fourth Power

Theorem

$\forall a, b, c \in \Z_{>0}$, the equation $a^4 + b^4 = c^4$ has no solutions.

Proof

This is a direct consequence of Fermat's Right Triangle Theorem.

$\blacksquare$

Historical Note

Pierre de Fermat gave an outline of the proof of Sum of two Fourth Powers cannot be Fourth Power using the Method of Infinite Descent.

It was proved in full by Leonhard Paul Euler.

It can of course be seen to be a special case of Fermat's Last Theorem.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs