Put $x = -t^4, y = t^3$ and $z = 0$ and see that it works. Hence infinite integer values satisfy the Equation

Choose $x = { 2 }^{ 4r }$ and $y = { 2 }^{ 3r }$. Then we will get $12r + 1 = 31k$ ( take $z = {2}^{k}$). And it is easy to prove that there exists infinitely many solutions in integers by applying Euclidean algorithm.

Doesn't the problem ask for prime triples?

aops29 wrote: Doesn't the problem ask for prime triples? I don't think the original problem does. Here's a generating triplet $(x,y,z)=(2^{124x+72},2^{93x+54},2^{12x+7})$

$-t^4,t^3$.