XXVI - I - Problem 5

Determine all integers $ m $ for which the polynomial $ x^3-mx^2+mx-(m^2+1) $ has an integer root.
Suppose that the integers $ x $ and $ m $ satisfy the equation $ x^3 - mx^2+mx-(m^2+1) = 0 $, which is $ (x^2+m)(x-m) = 1 $. Then $ x^2+m = 1 $ and $ x- m = 1 $ or $ x^2+m = -1 $ and $ x-m = -1 $. Adding the equations in each of these systems, we get $ x^2+x = 2 $ or $ x^2+x = - 2 $. The roots of the first equation are $ x = 1 $ and $ x = -2 $. They correspond to the numbers $ m = 0 $ and $ m = - 3 $. The second equation has no solutions in the set of integers.