Proof that $x^2 + D = AB^y$ has in every case of $D,A,B$ a finite amount of solutions $x,y$

Question

Could somebody please find me a proof that $$x^2 + D = AB^y$$ has in every case of $D,A,B$ a finite amount of solutions $x,y$. I forgot how this is called and would greatly appreciate it if someone could link me the proof

Answer 1

This is not true for $A = 0$ or $B = 1$: there are infinitely many $y$ that can satisfy the equation.