On the Largest Prime factor of the $k$-generalized Pell numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910920324677632 |
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| author | Batte, Herbert |
| author_facet | Batte, Herbert |
| contents | Let $k \ge 2$ be an integer and consider the $k$-generalized Pell sequence $\{P_n^{(k)}\}_{n \ge 2-k}$, defined by the initial values $0, \ldots, 0, 0, 1$ (a total of $k$ terms), and the recurrence $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$, for all $n\ge 2$. For any integer $m$, let $\mathcal{P}(m)$ denote the largest prime factor of $m$, with the convention $\mathcal{P}(0) = \mathcal{P}(\pm1) = 1$. In this paper, we prove that for $n \ge 4$, the inequality $\mathcal{P}(P_n^{(k)}) > (1/104) \log \log n$ holds. Additionally, we find all $k$-generalized Pell numbers $P_n^{(k)}$, whose largest prime factor does not exceed $7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Largest Prime factor of the $k$-generalized Pell numbers Batte, Herbert Number Theory 11B39, 11D61, 11D45, 11Y50 Let $k \ge 2$ be an integer and consider the $k$-generalized Pell sequence $\{P_n^{(k)}\}_{n \ge 2-k}$, defined by the initial values $0, \ldots, 0, 0, 1$ (a total of $k$ terms), and the recurrence $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$, for all $n\ge 2$. For any integer $m$, let $\mathcal{P}(m)$ denote the largest prime factor of $m$, with the convention $\mathcal{P}(0) = \mathcal{P}(\pm1) = 1$. In this paper, we prove that for $n \ge 4$, the inequality $\mathcal{P}(P_n^{(k)}) > (1/104) \log \log n$ holds. Additionally, we find all $k$-generalized Pell numbers $P_n^{(k)}$, whose largest prime factor does not exceed $7$. |
| title | On the Largest Prime factor of the $k$-generalized Pell numbers |
| topic | Number Theory 11B39, 11D61, 11D45, 11Y50 |
| url | https://arxiv.org/abs/2504.19052 |