Solution to the Diophantine equation $x^4+y^4=2z^2$ Does there exists a nontrivial positive integer solution with $x\ne y,$ of
$$x^4+y^4=2z^2.$$

First, I suppose that you know there no exist $x,y,z\in\Bbb Z^+$ such that
$$x^4-y^4=z^2.$$
You might have seen this while proving FLT for $n=4$ and further theorems. I can give you a hint for this if you need.
If $x^4+y^4=2z^2$, we have
$$z^4-(xy)^4=\left({x^4-y^4\over2}\right)^2.$$
Due to the above result, $z=0$ or $x=0$ or $y=0$ or $|x|=|y|$, and these are all trivial solutions.
Thus, there is no non-trivial solution.
