How to prove or disprove that $m^2 - n^2 = p$, where $p$ is odd, has exactly one positive solution if and only if $p$ is a prime $>2$.
Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.