Find the integral solutions to $ x^2+y^2+z^2=x^2y^2$

Question

I am unfamiliar with this type of problem. How does one solve this and under what category of math does this fall under.

Find the integral solutions for $x^2+y^2+z^2=x^2y^2$

Answer 1

$$ z^2 + 1 = (x^2 - 1)(y^2 - 1). $$

The main thing you need to know from quadratic forms is that $z^2 + 1$ is not divisible by $4$ or by any prime $q \equiv 3 \pmod 4,$ indeed not divisible by any (positive) number $n \equiv 3 \pmod 4.$

If either $x$ or $y$ is even and nonzero, let's say $x,$ then $x^2 - 1 \equiv 3 \pmod 4,$ not allowed.

When both $x,y$ are odd, both $x^2 -1$ and $y^2-1$ are even and $z^2+1$ is divisible by $4,$ again not allowed.

Therefore $(0,0,0)$ is it.