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Utilizing Symmetry in Finding New Permutiples from Known Examples

Benjamin V. Holt

TL;DR

The paper tackles the problem of finding new permutiples from known examples by exploiting symmetry in the mother graph and the Hoey-Sloane machine. It formalizes the combinatorial structures (graphs, cycle-images, and multigraphs) and develops symmetry-based techniques, including reflective and dihedral siblings, symmetric closures, and Eulerian circuits, to generate and classify additional permutiples with the same digit multisets. The results provide a unified framework that recasts prior constructions, enables systematic generation of new examples, and clarifies when distinct permutiples share the same graph or class. This symmetry-centered perspective enhances both the construction of new permutiples and the understanding of their structural relationships across bases and multipliers.

Abstract

A permutiple is a natural number whose representation in some base, $b>1$, is an integer multiple of a number whose base-$b$ representation has the same collection of digits. Previous efforts have made progress in finding such numbers using graph-theoretical and finite-state-machine constructions. These are the mother graph and the Hoey-Sloane machine. In this paper, we leverage the inherent symmetry of the above constructions for the purpose of finding new permutiples from old. Such results also help us to see previous work through a new lens.

Utilizing Symmetry in Finding New Permutiples from Known Examples

TL;DR

The paper tackles the problem of finding new permutiples from known examples by exploiting symmetry in the mother graph and the Hoey-Sloane machine. It formalizes the combinatorial structures (graphs, cycle-images, and multigraphs) and develops symmetry-based techniques, including reflective and dihedral siblings, symmetric closures, and Eulerian circuits, to generate and classify additional permutiples with the same digit multisets. The results provide a unified framework that recasts prior constructions, enables systematic generation of new examples, and clarifies when distinct permutiples share the same graph or class. This symmetry-centered perspective enhances both the construction of new permutiples and the understanding of their structural relationships across bases and multipliers.

Abstract

A permutiple is a natural number whose representation in some base, , is an integer multiple of a number whose base- representation has the same collection of digits. Previous efforts have made progress in finding such numbers using graph-theoretical and finite-state-machine constructions. These are the mother graph and the Hoey-Sloane machine. In this paper, we leverage the inherent symmetry of the above constructions for the purpose of finding new permutiples from old. Such results also help us to see previous work through a new lens.

Paper Structure

This paper contains 10 sections, 26 theorems, 50 equations, 19 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Basic definitions and results
  3. Permutiple graphs and the mother graph
  4. Finite-state-machine description of general digit-preserving multiplication and the Hoey-Sloane graph
  5. A multigraph representation of the Hoey-Sloane machine
  6. Reflective symmetry of the mother graph and Hoey-Sloane graph
  7. Symmetric classes
  8. New permutiples from old
  9. Other symmetries involving dihedral siblings
  10. Characterizing permutiple symmetries

Key Result

Theorem 1

Let $(d_k, d_{k-1},\ldots, d_0)_b$ be an $(n,b,\sigma)$-permutiple, and let $c_j$ be the $j^{th}$ carry. Then, $b c_{j+1}-c_j=nd_{\sigma(j)}-d_{j}$ for all $0\leq j \leq k$.

Figures (19)

  • Figure 1: The graph with edges $\{(d_j,d_{\tau(j)}) \mid 0 \leq j \leq 4 \}$ for each example in Table \ref{['conj_class_table']}.
  • Figure 2: The $(4,10)$-mother graph with the graph of $C$ displayed in bold red.
  • Figure 3: A labeled edge on the state diagram.
  • Figure 4: The $(4,10)$-Hoey-Sloane graph with cycle images of $G_C$ in bold red.
  • Figure 5: Two multi-edges representing two possible transitions from state $c_1$ to state $c_2$.
  • ...and 14 more figures

Theorems & Definitions (60)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Theorem 3
  • Definition 4
  • Theorem 4
  • Definition 5
  • Theorem 5
  • ...and 50 more