"Let $R$ be a ring of complex numbers. Assume $R$ is a UFD and has only finitely many units.Let $k$ be a non-zero element of $R$ Prove that;
$$x^2 - y^2 = k$$
Has only finitely many solutions with $x,y \in R$. Can the said be the same if we have infinitely many units?"
My first attempt was to split this equation up into 2 factors;
$$(x+y)(x-y)=k,$$
and then using the fact that R is a UFD to show that k has a unique factorization that we can exploit to find simultaneous equations in $x$ and $y$, but couldn't quite get the answer I wanted. I'm wondering what units have to do with it and how to use them to prove the statement?