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Permutation--invariant Niven numbers

Hui-Ling Wu, S. Y. Lou

TL;DR

The paper introduces permutation-invariant Niven numbers (PINNs), a new class of Harshad numbers where every digit permutation (ignoring leading zeros) yields another Niven number. It proves infinitude and unbounded growth of PINNs, shows zero natural density, and presents a two-stage algorithm to exhaustively generate PINNs for increasing digit lengths, culminating in a main theorem that for $k \ge 10$ all PINNs can be constructed from finite digit-blocks under $\mathcal{S}_k$ symmetry. The work provides explicit classifications for small lengths, extends to higher digits with a structured decomposition, and develops the mathematical framework (symmetry, digit-sum conditions, and modular congruences) to study PINNs while outlining open problems and future directions. Overall, PINNs blend modular arithmetic with digit-permutation symmetry to reveal infinite families, precise generation methods, and intriguing connections to repdigits and primes, with potential extensions to other bases and generalized Niven variants.

Abstract

This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation--invariant Niven numbers.

Permutation--invariant Niven numbers

TL;DR

The paper introduces permutation-invariant Niven numbers (PINNs), a new class of Harshad numbers where every digit permutation (ignoring leading zeros) yields another Niven number. It proves infinitude and unbounded growth of PINNs, shows zero natural density, and presents a two-stage algorithm to exhaustively generate PINNs for increasing digit lengths, culminating in a main theorem that for all PINNs can be constructed from finite digit-blocks under symmetry. The work provides explicit classifications for small lengths, extends to higher digits with a structured decomposition, and develops the mathematical framework (symmetry, digit-sum conditions, and modular congruences) to study PINNs while outlining open problems and future directions. Overall, PINNs blend modular arithmetic with digit-permutation symmetry to reveal infinite families, precise generation methods, and intriguing connections to repdigits and primes, with potential extensions to other bases and generalized Niven variants.

Abstract

This paper introduces permutation-invariant Niven numbers--a novel class of Niven numbers where all digit permutations (with leading zeros automatically ignored) must retain the Niven property. We demonstrate that there exist infinitely many such numbers and that their magnitude is unbounded. Furthermore, we present an exhaustive search method for identifying permutation--invariant Niven numbers.

Paper Structure

This paper contains 22 sections, 1 theorem, 61 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Research Questions
  4. Basic Properties
  5. Single-digit PINNs
  6. Repdigits as PINNs
  7. Numbers Containing Zero
  8. Infinitude and Density
  9. Exhaustive Search for Small PINNs
  10. 2-digit PINNs
  11. 3-digit PINNs
  12. Algorithmic Generation of PINNs
  13. 4-digit PINNs
  14. Higher-digit PINNs
  15. Main Theorem
  16. ...and 7 more sections

Key Result

Theorem 1

The $k$--digit integers with $k\geq10$ are all PINNs, where ${\cal{N}}_{ka},\ {\cal{N}}_{kb},\ {\cal{N}}_{kc},\ {\cal{N}}_{kd},\ {\cal{N}}_{ke},\ {\cal{N}}_{kf},\ {\cal{N}}_{kg},\ {\cal{N}}_{kh},\ {\cal{N}}_{ki},$ and ${\cal{N}}_{kj}$ are defined by with the notations $k_i\equiv (k-i),\ i=1,\ \ldots, 9$.

Theorems & Definitions (3)

  • Definition 1
  • Conjecture 1
  • Theorem 1