Powerful Fibonacci polynomials over finite fields

TL;DR

This paper characterizes Fibonacci and related polynomial sequences over finite fields that are perfect powers or powerful, extending classical number theory results to polynomial analogues with infinitely many such sequences.

Contribution

It provides a complete characterization of Fibonacci polynomials that are perfect powers over finite fields, and extends the analysis to Horadam's generalized Lucas polynomial sequences.

Findings

Fibonacci polynomials that are perfect powers over finite fields are fully characterized.

Infinitely many Fibonacci and related polynomials are powerful over finite fields.

The results include characterizations for Lucas, Chebyshev, and Jacobsthal polynomial sequences.

Abstract

Bugeaud, Mignotte, and Siksek proved that the only perfect powers in Fibonacci sequence are 0, 1, 8, and 144. In this paper, we study the polynomial analogue of the problem. Especially, we give a complete characterization of the Fibonacci polynomials that are perfect powers or powerful over finite fields, where there are infinitely many of them. We also give similar characterizations for some of Horadam's generalized Lucas polynomial sequences, which include Fibonacci, Lucas, Chebyshev, and Jacobsthal polynomials.