Representing an integer and its powers in two unrelated number systems
Volume 223 / 2026
Acta Arithmetica 223 (2026), 309-338
MSC: Primary 11D61; Secondary 11J86, 11A63, 11B39
DOI: 10.4064/aa250218-13-4
Published online: 27 May 2026
Abstract
We prove that under a necessary independence assumption, there are only finitely many integers having bounded sum-of-digits in base $b_1$ while also having some integral power with bounded sum-of-digits in base $b_2$ where $b_1$ and $b_2$ are coprime. These powers can also be effectively listed in principle. Furthermore, analogous results involving both radix and Ostrowski numeration systems have been established.
