I’ve done some numerical experiments with different values of , but have yet to see patterns or generate conjectures.

Poking around Sloane’s On-line Encyclopedia of Integer Sequences for related sequences (values of where solutions exist, or of *c*, or etc.) came up empty.

Dickson’s *History of the Theory of Numbers* mentioned one related result involving triangular numbers, but I want to track down the source before commenting further on that. (And Dickson’s *History…* is huge — I’m not confident that I didn’t miss something else there.)