Find all integers n for which the equation x³ - 13x + n = 0 has three integer roots.

Since the x² coefficient is 0, the three roots are p, q, and -(p+q). From the relationship between roots and coefficients, we also know that p.q.(p+q)=n, and pq-p(p+q)-q(p+q)=-13, so p²+pq+q²=-13. The last one is equivalent to (p³-q³)/(p-q)=-13... but at first sight I don't see if this can have integer solutions.

Posted by Federico Kereki on 2006-10-14 16:52:52