Sums of S-units in X-coordinates of Pell equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912122772914176 |
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| author | Nair, Parvathi S Rout, Sudhansu Sekhar |
| author_facet | Nair, Parvathi S Rout, Sudhansu Sekhar |
| contents | Let $S$ be a fixed set of primes and let $(X_{l})_{l\geq 1}$ be the $X$-coordinates of the positive integer solutions $(X, Y)$ of the Pell equation $X^2-dY^2 = 1$ corresponding to a non-square integer $d>1$. We show that there are only a finite number of non-square integers $d>1$ such that there are at least two different elements of the sequence $(X_{l})_{l\geq 1}$ that can be represented as a sum of $S$-units with a fixed number of terms. Furthermore, we solve explicitly a particular case in which two of the $X$-coordinates are product of power of two and power of three. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sums of S-units in X-coordinates of Pell equations Nair, Parvathi S Rout, Sudhansu Sekhar Number Theory 11B37 (Primary) 11D45, 11J86 (Secondary) Let $S$ be a fixed set of primes and let $(X_{l})_{l\geq 1}$ be the $X$-coordinates of the positive integer solutions $(X, Y)$ of the Pell equation $X^2-dY^2 = 1$ corresponding to a non-square integer $d>1$. We show that there are only a finite number of non-square integers $d>1$ such that there are at least two different elements of the sequence $(X_{l})_{l\geq 1}$ that can be represented as a sum of $S$-units with a fixed number of terms. Furthermore, we solve explicitly a particular case in which two of the $X$-coordinates are product of power of two and power of three. |
| title | Sums of S-units in X-coordinates of Pell equations |
| topic | Number Theory 11B37 (Primary) 11D45, 11J86 (Secondary) |
| url | https://arxiv.org/abs/2411.11103 |