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Powerful Fibonacci polynomials over finite fields

Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim

TL;DR

This work addresses Fibonacci polynomials over finite fields, delivering complete classifications for when these polynomials are perfect $j$-th powers and when they are powerful. The authors use a polynomial Binet-type framework, explicit finite-field factorization $F_{p^k m}(T)=F_{p^k}(T)F_m(T)^{p^k}$, and discriminant-based square-free arguments to reduce the problem to $n = p^{ak}$ and $m=1$, yielding a precise condition $n = p^{a k}$ with $a = \mathrm{ord}_{2j}(p)$ (and separate treatment for $p=2$). They also provide analogous results for Horadam’s Lucas-type polynomials, demonstrating parallel power/powerful characterizations and explicit factorizations; these results reveal an infinite abundance of polynomial perfect powers over finite fields. The work combines the polynomial analogue of Binet’s formula with discriminant theory to extend classical integer results to the polynomial and finite-field setting, offering new insights into the arithmetic of Fibonacci-type polynomials.

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.

Powerful Fibonacci polynomials over finite fields

TL;DR

This work addresses Fibonacci polynomials over finite fields, delivering complete classifications for when these polynomials are perfect -th powers and when they are powerful. The authors use a polynomial Binet-type framework, explicit finite-field factorization , and discriminant-based square-free arguments to reduce the problem to and , yielding a precise condition with (and separate treatment for ). They also provide analogous results for Horadam’s Lucas-type polynomials, demonstrating parallel power/powerful characterizations and explicit factorizations; these results reveal an infinite abundance of polynomial perfect powers over finite fields. The work combines the polynomial analogue of Binet’s formula with discriminant theory to extend classical integer results to the polynomial and finite-field setting, offering new insights into the arithmetic of Fibonacci-type polynomials.

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.
Paper Structure (10 sections, 32 theorems, 38 equations, 5 tables)

This paper contains 10 sections, 32 theorems, 38 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diophantine equations with Fibonacci numbers
  4. Fibonacci polynomials
  5. Lucas polynomial sequence
  6. Proof of Theorem \ref{['thm:mainFn']}
  7. Factorization over finite fields
  8. Square-free factor
  9. Proof of the main theorem
  10. Generalization

Key Result

Theorem 1.1

Let $j > 1$. Let $p$ be an odd prime number coprime to $2j$, $q$ be a power of $p$, and let $a = \mathrm{ord}_{2j}(p)$ denote the order of $p$ in $\left( \mathbb{Z}/2j\mathbb{Z} \right)^\times$. For $n > 0$, $F_n(T)$ is a perfect $j$-th power in $\mathbb{F}_q[T]$ if and only if $n = p^{ak}$ for some

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: bugeaud2006classical
  • Proposition 2.2: kitayama2017irreducibility
  • Definition 2.3: Horadam1996
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 51 more