On the resolution of the Diophantine equation $U_n + U_m = x^q$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917733503860736 |
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| author | Bhoi, P. K. Rout, S. S. Panda, G. K. |
| author_facet | Bhoi, P. K. Rout, S. S. Panda, G. K. |
| contents | Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $α$ with $α>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11934 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the resolution of the Diophantine equation $U_n + U_m = x^q$ Bhoi, P. K. Rout, S. S. Panda, G. K. Number Theory 11B39, 11D61, 11J86 Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $α$ with $α>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation. |
| title | On the resolution of the Diophantine equation $U_n + U_m = x^q$ |
| topic | Number Theory 11B39, 11D61, 11J86 |
| url | https://arxiv.org/abs/2202.11934 |