Sums of Fourth Powers While fooling around on my calculator I found:
$$7^4 + 8^4 + (7 + 8)^4 = 2 * 13^4$$
$$11^4 + 24^4 + (11 + 24)^4 = 2 * 31^4$$
I'm intrigued but I can't explain why these two equations are true. Are these coincidences or is there a formula/theorem explaining them?

I'll start at
Will Jagy's hint.
If
$c^2
=a^2+ab+b^2
$,
$\begin{array}\\
c^4
&=a^4+a^2b^2+b^4+2a^3b+2a^2b^2+2ab^3\\
&=a^4+3a^2b^2+b^4+2a^3b+2ab^3\\
\text{so}\\
2c^4
&=2a^4+6a^2b^2+2b^4+4a^3b+4ab^3\\
&=a^4+b^2+a^4+6a^2b^2+b^4+4a^3b+4ab^3\\
&=a^4+b^4+a^4+4a^3b+6a^2b^2+4ab^3+b^4\\
&=a^4+b^4+(a+b)^4\\
\end{array}
$
Yep.
