On the difference between squares and integral $S$-units
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915254658662400 |
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| author | Bugeaud, Yann |
| author_facet | Bugeaud, Yann |
| contents | Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers and $x$ a positive integer. We establish an effective lower bound for the greatest prime divisor of $|x^2 - q_1^{a_1} \ldots q_t^{a_t}|$, which tends to infinity with the maximum of $x$, $a_1, \ldots , a_t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the difference between squares and integral $S$-units Bugeaud, Yann Number Theory 11D61, 11J86 Let $q_1, \ldots , q_t$ be distinct prime numbers. Let $a_1, \ldots , a_t$ be nonnegative integers and $x$ a positive integer. We establish an effective lower bound for the greatest prime divisor of $|x^2 - q_1^{a_1} \ldots q_t^{a_t}|$, which tends to infinity with the maximum of $x$, $a_1, \ldots , a_t$. |
| title | On the difference between squares and integral $S$-units |
| topic | Number Theory 11D61, 11J86 |
| url | https://arxiv.org/abs/2503.22084 |