Determine all possible pairs (P,Q) of positive integers such that each of the roots of:
X^{3}  17X^{2} + Px  Q^{2} = 0 is an integer.
(In reply to
solution by xdog)
Ok, xdog. I made a sign mistake, so I didn't search for all+ roots.
P Q must be positive integers, but it does not imply that all the roots should necessarily be positive.
With one negative root either: or the sign of P changes as in (18, 2, 1) 20 36, or the sign of Q^2 changes as in (15, 5, 3) 15, 15.
A solution with two negative roots and one positive need consideration before ruling it out. But in fact leads to impossibility.
Edited on April 24, 2016, 12:41 pm

Posted by armando
on 20160424 11:16:20 