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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.11252 |
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| _version_ | 1866908589362249728 |
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| author | Yamada, Tomohiro |
| author_facet | Yamada, Tomohiro |
| contents | We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11252 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function Yamada, Tomohiro Number Theory 11D61, 11A05, 11D45, 11P21 We prove that the sum of reciprocals $1/x$ of integer solutions of $(x^m-1)/(x-1)=N$ with $x, m\geq 2$ for a given integer $N$ except the smallest $x$ is smaller than $5.9037$. If we limit $x$ to be prime, then the sum is smaller than $0.73194$. |
| title | On the Ratat-Goormaghtigh equation and integer points close to the graph of a smooth function |
| topic | Number Theory 11D61, 11A05, 11D45, 11P21 |
| url | https://arxiv.org/abs/2510.11252 |