Solve equation with two unknowns (maybe modulo)

Question

Given the following equation: $$ x^{2} - y^{2}=17, \quad 0\neq x,y\in \mathbb N$$

I know for example that one solution is $x=9$, $y=8$, but I do not know how to get it.

Answer 1

rewrite your equation in the form $(x-y)(x+y)=1\cdot 17$ from here you will get $x-y=1$ and $x+y=17$ the other case is impossible. or we have $(x-y)(x+y)=(-1)\cdot (-17)$