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?

You have a disguised version of triangles with integer sides and one $120^\circ$ angle. These are
$$ 3,5,7 $$
$$ 7,8,13 $$
$$ 5,16,19$$
$$ 11,24, 31,  $$
which solve
$$ a^2 + ab + b^2 = c^2. $$
Square both sides and then double both sides and you get your identities.
These can be generated by a coprime pair of number $m,n$ with
$$ a = m^2 - n^2 $$
$$ b = 2mn+n^2  $$
$$ c = m^2 + mn + n^2 $$
