Number of integer solutions $(x, y)$ of $x(x+6) = y^2 + k$ for different integer values of $k$

Question

Let $n$ be the number of pairs $(x, y)$ of integer solutions to the following equation:$$x(x+6) = y^2 + k$$

Can there be an integer $m$, $k$ can be given an integer value so that $n=m$ ?

Answer 1

A hint: Put $z=x+3$ and simplify to an equation containing $z^2-y^2$. You can then apply a well-known analysis of differences between squares equal to a given integer, as in answers to this question.