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arXiv · 2602.01252

Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

Scott Duke Kominers
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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 $\ell=1,\ldots, k$).

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Scott Duke Kominers. 2026-02-01. Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases. https://arxiv.org/abs/2602.01252

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