Determine all possible pairs (P,Q) of positive integers such that each of the roots of:
X3 - 17X2 + Px - Q2 = 0 is an integer.
(In reply to solution
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 2016-04-24 11:16:20