Arithmetic progressions in sumsets of geometric progressions

2025/12/03 by Bennett, Michael A.
#11D61 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.03498

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, a², a³, … and 1, b, b², b³, …. 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.

Citations

• On the length of an arithmetic progression of the form 3^x+2^y
• Effective S-unit Equations Beyond 3 Terms : Newman's Conjecture

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