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arXiv · 2601.02664

Powerful Fibonacci polynomials over finite fields

Graeme Bates·Ryan Jesubalan·Seewoo Lee·Jane Lu·Hyewon Shim
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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.

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Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim. 2026-01-06. Powerful Fibonacci polynomials over finite fields. https://arxiv.org/abs/2601.02664

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