On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929394830802944 |
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| author | Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac |
| author_facet | Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac |
| contents | Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and $P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m, n, k, b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le 10.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_11607 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers Adédji, Kouèssi Norbert Trebješanin, Marija Bliznac Number Theory 11B39, 11J68, 11J86 Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and $P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in base $b,$ have only finitely many solutions in nonegative integers $(m, n, k, b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le 10.$ |
| title | On Mixed Concatenations of Pell and Pell-Lucas Numbers which are Pell Numbers |
| topic | Number Theory 11B39, 11J68, 11J86 |
| url | https://arxiv.org/abs/2304.11607 |