Solve the equation:$$x^4+y^4=d*z^2,$$ where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer.
I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is solvable,so $p\equiv 1 \pmod 8$,but this is not sufficient.
PS:I just want to know how to determine whether $x^4+y^4=d*z^2$ has any positive integer solutions,not ask how to get all the solutions.
Thanks in advance!