Determine all possible pairs (p,q) of positive integers such that :
pq²+ q + 23 divides p²q + p + q .

1. Prove that no solution exists for p< q; so p> = q.
2. Substitute s = ((p^2)*q + p + q) / (p*(q^2) + q + 23)
3. This would  give a simple parametric solution for x and y in terms of s in a particular situation. In all other available  situations q must be less than or equal to 4.(Why?)
4. It is now relatively easy to determine all possible solutions to the problem.

Edited on June 8, 2006, 11:27 pm

Posted by K Sengupta on 2006-06-08 23:26:37