Arithmetic progressions in sumsets of geometric progressions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918229724626944 |
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| author | Bennett, Michael A. |
| author_facet | Bennett, Michael A. |
| contents | If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications of bounds for linear forms in logarithms to $S$-unit equations, and consequences of the modularity of Frey-Hellegouarch curves, together with elementary arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03498 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic progressions in sumsets of geometric progressions Bennett, Michael A. Number Theory 11D61 If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications of bounds for linear forms in logarithms to $S$-unit equations, and consequences of the modularity of Frey-Hellegouarch curves, together with elementary arguments. |
| title | Arithmetic progressions in sumsets of geometric progressions |
| topic | Number Theory 11D61 |
| url | https://arxiv.org/abs/2512.03498 |