For some nonzero integer N, the equation: X³- 2017X+N = 0 has the three integer roots p, q, and r

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

A graph of Y=X³- 2017X shows the characteristic cubic form.  N will just shift it up or down.  What we seek is the value of N that will make all solutions integers.

This is equivalent to finding three values of X in the above equation that give the same value of Y.  From the graph, these must be in the interval [-51,51].

What I did was make a take of X,Y and then sorted by Y.  A quick scan found only one triple repeat of Y. 
(-48,-13776), (7,-13776), (41,-13776)
(There were also no duplicates.)

This implies the solution 48+7+41=96 and also
X³- 2017X+13776=(X+48)(X-7)(X-41)

Posted by Jer on 2016-06-22 13:37:44