Reducible polynomials 2

Algebra Level pending

Find the smallest positive integer $n$ such that $x^{17}-x+n$ is reducible over the rationals (i.e., it can be factored into at least two nontrivial rational factors).

The answer is 131070.

1 solution

Christopher Criscitiello
Dec 2, 2017

Hint: It is not difficult to show that $x^{17}-x+131070$ is reducible (see https://brilliant.org/problems/more-polynomials-2/?ref_id=1436779 ). The difficulty is in showing this is the least possible $n$ .