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Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

Scott Duke Kominers

TL;DR

The paper proves that for any base $b\ge2$, any $k\ge1$ (with $B=b^k$), and any arithmetic progression $S_{m,r}$ with $\gcd(m,b)=1$, there exist infinitely many integers in $S_{m,r}$ that are simultaneously $b$-Niven and $B$-Niven, and moreover one can achieve $s_b(n)=s_B(n)\ge s_0$ for any prescribed $s_0$. The core method combines a sparse repunit construction in base $B$ with a digit-sum compatibility observation: when base-$B$ digits are restricted to $\{0,1,\ldots,b-1\}$, the base-$b$ digit sum matches the base-$B$ digit sum, preserving no carries. By choosing the spacing through the multiplicative order $\text{ord}_{ms}(B)$ and using the CRT to place $n$ in the desired residue class mod $m$, the authors force both the progression condition and divisibility by the digit sum. The results extend to simultaneous Niven-ism across multiple power-related bases and suggest further extensions to sets of bases beyond the power-related framework. This provides an explicit, constructive infinite family of numbers with dual Niven properties in structured arithmetic progressions, with potential implications for related base-interaction problems in number theory.

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 $\ell=1,\ldots, k$).

Simultaneous Niven Numbers in Arithmetic Progressions for Power-Related Bases

TL;DR

The paper proves that for any base , any (with ), and any arithmetic progression with , there exist infinitely many integers in that are simultaneously -Niven and -Niven, and moreover one can achieve for any prescribed . The core method combines a sparse repunit construction in base with a digit-sum compatibility observation: when base- digits are restricted to , the base- digit sum matches the base- digit sum, preserving no carries. By choosing the spacing through the multiplicative order and using the CRT to place in the desired residue class mod , the authors force both the progression condition and divisibility by the digit sum. The results extend to simultaneous Niven-ism across multiple power-related bases and suggest further extensions to sets of bases beyond the power-related framework. This provides an explicit, constructive infinite family of numbers with dual Niven properties in structured arithmetic progressions, with potential implications for related base-interaction problems in number theory.

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- Niven numbers, for any fixed . 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 contains infinitely many integers that are simultaneously -Niven and -Niven (indeed, we can obtain simultaneous -Niven-ness for ).
Paper Structure (14 sections, 6 theorems, 35 equations)

This paper contains 14 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Digit sums and base expansions
  5. Compatibility between bases $b$ and $B=b^k$
  6. Multiplicative order
  7. The construction
  8. Verification of the desired properties
  9. Infinitude of the admissible $s$
  10. An illustrative example
  11. Concluding remarks
  12. Relation to the order--repunit method in the coprime case
  13. Extension to multiple power-related bases
  14. Potential for further extensions

Key Result

Theorem 1.1

Let $b \ge 2$ and $k \ge 1$ be integers, and set $B = b^k$. If $m\ge 1$ is such that $\gcd(m,b)=1$, then for any $r$ with $0\le r<m$, there exist infinitely many integers $n \in S_{m,r}$ such that i.e., there are infinitely many $n$ in the arithmetic progression $S_{m,r}$ that are simultaneously $b$- and $b^k$-Niven. Moreover, for any $s_0\geq1$, there exists such an $n$ with $\mathsf{s}_b(n) =\m

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 4 more