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  • $\begingroup$ I suspect the reason you only see $c \in \{\pm 1, \pm 2, \pm 4\}$ is because quadratic forms happen to have degree $2$. Also, for general $c$ this equation will boil down to the number of factorizations of $c$, and the class number of the quadratic field $\mathbb{Q}(\sqrt{ab})$ inevitably creeps in. $\endgroup$ Commented Feb 1 at 17:24
  • $\begingroup$ artofproblemsolving.com/community/c3046h2938816 $\endgroup$ Commented Feb 1 at 18:16
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    $\begingroup$ @individ I do not think this helps at all - it only gives some family of solutions, but in general these are even not all solutions to the equation. The question here addresses a statement which needs to be valid for all solutions. $\endgroup$ Commented Feb 2 at 0:48