Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

TL;DR

This paper extends the study of Niven numbers in arithmetic progressions by demonstrating that, for bases and differences coprime to the base, there are infinitely many numbers that are simultaneously Niven in multiple related bases, using a novel sparse repunit approach.

Contribution

It introduces a structured two-base approach to find infinitely many numbers that are simultaneously Niven in multiple bases within arithmetic progressions.

Findings

Proves existence of infinitely many simultaneous Niven numbers in specific bases.

Uses a sparse repunit construction to achieve the result.

Extends previous work on Niven numbers in arithmetic progressions.

Abstract

Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to $b$ contains infinitely many integers that are simultaneously $b$-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell$-Niven-ness for $\ell=1,\ldots, k$).