Determine all possible integer pairs (p,q) such that p+q²+s³=pqs, where s=gcd(p,q) and gcd denotes the

greatest common divisor.
By a process similiar to that of my previous post, we can arrive at the following:

(s^2)*p = (sq + 1) + (s^5 + 1)/(sq - 1)

A necessary (but not sufficient) condition for p and q to be integers is that (sq - 1) must divide (s^5 + 1).

For instance, if s = 3, (3q - 1) must divide 244.

244 = 2 * 2 * 61, so 3q - 1 must = 1 or 2 or 4 or 61 or 122 or 242 (if q is positive). The only positive integral q's consistent with this are q = 43, and this is not divisible by the required gcd of 3.

In general, (s^5 + 1) must have a divisor which is a*s^2 + 1, where a is integral. And even this is not sufficient ...