Powerful Fibonacci polynomials over finite fields
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917186350612480 |
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| author | Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon |
| author_facet | Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02664 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Powerful Fibonacci polynomials over finite fields Bates, Graeme Jesubalan, Ryan Lee, Seewoo Lu, Jane Shim, Hyewon Number Theory 11T06 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. |
| title | Powerful Fibonacci polynomials over finite fields |
| topic | Number Theory 11T06 |
| url | https://arxiv.org/abs/2601.02664 |