 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Three Integer Root Sum (Posted on 2016-06-22) 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 No Rating Comments: ( Back to comment list | You must be logged in to post comments.) 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 Please log in:

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