Zeros and $S$-units in sums of terms of recurrence sequences in function fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929704817131520 |
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| author | N, Darsana Rout, S. S. |
| author_facet | N, Darsana Rout, S. S. |
| contents | Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_\ell = 0$ in $(n, m, \ell)\in \N^3$, where $U_n,V_m,W_\ell$ are simple linear recurrence sequences in the function field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeros and $S$-units in sums of terms of recurrence sequences in function fields N, Darsana Rout, S. S. Number Theory 11B37 (Primary), 11D61, 11R58 (Secondary) Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two defined over a function field and $\mathcal{O}_S^*$ be the set of $S$-units. In this paper, we use a result of Brownawell and Masser to prove effective results related to the Diophantine equations concerning linear recurrence sequences and $S$-units. In particular, we provide a finiteness result for the solutions of the Diophantine equation $U_{n_1} + \cdots + U_{n_r} \in \mathcal{O}_S^*$ in nonnegative integers $n_1, \ldots, n_r$. Furthermore, we study the finiteness result of the Diophantine equation $U_n+V_m+W_\ell = 0$ in $(n, m, \ell)\in \N^3$, where $U_n,V_m,W_\ell$ are simple linear recurrence sequences in the function field. |
| title | Zeros and $S$-units in sums of terms of recurrence sequences in function fields |
| topic | Number Theory 11B37 (Primary), 11D61, 11R58 (Secondary) |
| url | https://arxiv.org/abs/2408.09448 |