On Pell numbers representable as product of two generalized Fibonacci numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913948583854080 |
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| author | Bravo, Jhon J. Das, Pranabesh Herrera, Jose L. Saunders, John C. |
| author_facet | Bravo, Jhon J. Das, Pranabesh Herrera, Jose L. Saunders, John C. |
| contents | A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we find all Pell numbers that can be written as a product of two $k$-Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a variation of a result of Dujella and Pethő in Diophantine approximation. This work generalizes a prior result of Alekseyev which dealt with determining the intersection of the Fibonacci and Pell sequences, a work of Ddamulira, Luca and Rakotomalala who searched for Pell numbers which are products of two Fibonacci numbers, and a result of Bravo, Gómez, and Herrera, who found all Pell numbers appearing in the $k$-Fibonacci sequence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Pell numbers representable as product of two generalized Fibonacci numbers Bravo, Jhon J. Das, Pranabesh Herrera, Jose L. Saunders, John C. Number Theory 11B39, 11J86 A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms. In this paper, we find all Pell numbers that can be written as a product of two $k$-Fibonacci numbers. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a variation of a result of Dujella and Pethő in Diophantine approximation. This work generalizes a prior result of Alekseyev which dealt with determining the intersection of the Fibonacci and Pell sequences, a work of Ddamulira, Luca and Rakotomalala who searched for Pell numbers which are products of two Fibonacci numbers, and a result of Bravo, Gómez, and Herrera, who found all Pell numbers appearing in the $k$-Fibonacci sequence. |
| title | On Pell numbers representable as product of two generalized Fibonacci numbers |
| topic | Number Theory 11B39, 11J86 |
| url | https://arxiv.org/abs/2507.13674 |