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Integer Root Illation (Posted on 2016-04-23) Difficulty: 3 of 5
Determine all possible pairs (P,Q) of positive integers such that each of the roots of:
X3 - 17X2 + Px - Q2 = 0 is an integer.

No Solution Yet Submitted by K Sengupta    
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re: solution | Comment 3 of 5 |
(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 2016-04-24 11:16:20

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