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arXiv · 2512.03498

Arithmetic progressions in sumsets of geometric progressions

Michael A. Bennett
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Abstract

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.

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Michael A. Bennett. 2025-12-03. Arithmetic progressions in sumsets of geometric progressions. https://arxiv.org/abs/2512.03498

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