On the difference between perfect powers and integral $S$-units
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914520145854464 |
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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. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with $z^d$, where $z$ is a positive integer coprime with $q_1 \ldots q_t$ and $d \ge 2$ is an integer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the difference between perfect powers 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. We establish effective lower bounds for $|z^d - q_1^{a_1} \ldots q_t^{a_t}|$ and for its greatest prime factor, which tend to infinity with $z^d$, where $z$ is a positive integer coprime with $q_1 \ldots q_t$ and $d \ge 2$ is an integer. |
| title | On the difference between perfect powers and integral $S$-units |
| topic | Number Theory 11D61, 11J86 |
| url | https://arxiv.org/abs/2604.27490 |