TITLE: On the Diophantine equation $x^{4}+y^{4}=z^p$
QUESTION [5 upvotes]: Do there exist integers $x,y,z$ with $xyz\neq 0$, such that
$$x^4 + y^4 = z^p$$
where $p\geq 5$ is some prime ?
If yes, are there infinitely many of them ? And if there exists infinitely many of them, what are their parametrizations ?

REPLY [7 votes]: There are no primitive solutions, even when one of the fourth powers is replaced by a square. This was proved by Bennett, Ellenberg, Ng (see Int. J. Number Theory 6 (2010), 311-338). 
Non-primitive solutions there are plenty, see Noam Elkies's comment for an example.
P.S. I found the paper online here, I don't know how long the link lasts.