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Three Integer Root Sum (Posted on 2016-06-22) Difficulty: 3 of 5
For some nonzero integer N, the equation: X3- 2017X+N = 0 has the three integer roots p, q, and r

Find |p| + |q| + |r|.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 2
Exactly two of p,q,r have the same sign.  Let p and q be those roots.  

The quadriatic coefficient is 0, then r = -p-q.
The linear coefficient is -2017, which equals p*q+p*r+q*r.  This simplifies to 2017 = p^2 + p*q + q^2.  Since p and q are the same sign and integers, there are a finite set of candidates.  Specifically both p and q are in the range [1,45], each solution in that range has a mirror counterpart in [-45,-1].

A quick computer search finds one distinct solution: p=41, q=7.  Then |p|+|q|+|r| = 96.

  Posted by Brian Smith on 2016-06-22 10:17:01
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