Find positive integer solutions of the Diophantine equation $x^4+py^4=z^2$.

Question

I'm Trying to find all integer solutions of the diophantine equation:

$$x^4+py^4=z^2,$$

where $p$ is a prime number $p\equiv 13 \quad or\quad 17 \quad (\mod 20)$.

I know that $y=0$ is a solution of this. I think this is the only solution, but I can't prove this fact.

Can you help me to prove this claim if it's true, and to find all solutions otherwise.

Answer 1

The claim is FALSE.

For $p=13$, $x=3, y=2$ gives $3^4+13\times2^4=17^2$

For $p=37$, $x=151, y=30$ gives $151^4+37\times30^4=23449^2$