Find all integer numbers such that
Well, here's three for you for , none of which are positive though: . I'm not finding any more by blind searching. By making the substitutions , you can get a quadratic on : as long as . Solve for using quadratic formula and replace back in terms of and you can simplify to . Not sure if that's any better, but you might be able to parametrize that bit under the radical to get a list of pairs that give you a perfect square. That's as far as I've gotten so far. Was there a context for this?
There are exactly 4 integer solutions to this eqaution as calculated by WolframAlpha search engine ... here's the result:
(x^2+y)*(x+y^2) ;=(x-y)^2 - Wolfram|Alpha