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A Multigraph Characterization of Permutiple Strings

Benjamin V. Holt

TL;DR

The paper develops a multigraph framework to determine when base-b numbers with a given multiplier n form permutiples. By coupling the Hoey-Sloane machine with cycle multi-images on the mother graph, it proves that a multiset union of mother-graph cycles yields permutiple strings if and only if the corresponding cycle multi-images form an L-Eulerian multigraph. This yields a concrete, Eulerian-circuit-based method to construct permutiples of arbitrary length and base, and reduces counting such numbers to analyzing Eulerian circuits via established algorithms. The work also outlines future research directions, including palintiple cases and derived-permutiple phenomena, and discusses computational challenges as the base and multiplier grow.

Abstract

A permutiple is a natural number whose representation in some base is an integer multiple of a number whose representation has the same collection of digits. A previous paper utilizes a finite-state-machine construction and its state graph to recognize permutiples and to generate new examples. Permutiples are associated with walks on the state graph which necessarily satisfy certain conditions. However, the above effort does not provide sufficient conditions for the existence of permutiples. In this paper, we provide such a condition which we will state using the language of multigraphs.

A Multigraph Characterization of Permutiple Strings

TL;DR

The paper develops a multigraph framework to determine when base-b numbers with a given multiplier n form permutiples. By coupling the Hoey-Sloane machine with cycle multi-images on the mother graph, it proves that a multiset union of mother-graph cycles yields permutiple strings if and only if the corresponding cycle multi-images form an L-Eulerian multigraph. This yields a concrete, Eulerian-circuit-based method to construct permutiples of arbitrary length and base, and reduces counting such numbers to analyzing Eulerian circuits via established algorithms. The work also outlines future research directions, including palintiple cases and derived-permutiple phenomena, and discusses computational challenges as the base and multiplier grow.

Abstract

A permutiple is a natural number whose representation in some base is an integer multiple of a number whose representation has the same collection of digits. A previous paper utilizes a finite-state-machine construction and its state graph to recognize permutiples and to generate new examples. Permutiples are associated with walks on the state graph which necessarily satisfy certain conditions. However, the above effort does not provide sufficient conditions for the existence of permutiples. In this paper, we provide such a condition which we will state using the language of multigraphs.
Paper Structure (8 sections, 9 theorems, 7 equations, 13 figures, 7 tables)

This paper contains 8 sections, 9 theorems, 7 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Summary of Previous Work
  3. Basic Definitions and Results
  4. Permutiple Graphs and the Mother Graph
  5. Finite-State-Machine Description of the Permutiple Problem
  6. The Hoey-Sloane Multigraph
  7. A Multigraph Condition for Permutiple Strings
  8. Summary, Conclusions, and Future Work

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, for all $0\leq j \leq k$.

Figures (13)

  • Figure 1: The directed graph which results from taking the collection of ordered pairs $\left\{\left (d_j,d_{\sigma(j)}\right) |\, 0 \leq j \leq 4 \right\}$ from any example in Table \ref{['conj_class_table']} as edges.
  • Figure 2: The $(4,10)$-mother graph with the graph of $C$ featured in bold red.
  • Figure 3: An edge on the state diagram.
  • Figure 4: The $(4,10)$-Hoey-Sloane graph with cycle images of $G_C$ in bold red.
  • Figure 5: The $(2,4)$-mother graph.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Definition 1: holt_3
  • Theorem 1: holt_3
  • Theorem 2: holt_3
  • Definition 2: holt_5
  • Definition 3: holt_5
  • Theorem 3: holt_5
  • Definition 4: holt_5
  • Theorem 4: holt_5
  • Example 1
  • Corollary 1: holt_5
  • ...and 16 more