Polynomial values in Fibonacci sequences

Abstract

The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values $g\left(\mathbb{Z}\right)$ for any polynomial $g$ which is not equivalent to a Dickson polynomial.