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Finding Permutiples of a Known Base and Multiplier

Benjamin V. Holt

Abstract

Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of their digits (palintiples). Previous efforts have produced methods which construct new examples of permutiples with the same set of digits as a known example. Using simple graph-theoretical and finite-state machine constructions, we advance previous work by describing two methods for finding permutiples of a known base and multiplier with no need for known examples or prior knowledge of digits.

Finding Permutiples of a Known Base and Multiplier

Abstract

Natural numbers which are nontrivial multiples of some permutation of their base- digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of their digits (palintiples). Previous efforts have produced methods which construct new examples of permutiples with the same set of digits as a known example. Using simple graph-theoretical and finite-state machine constructions, we advance previous work by describing two methods for finding permutiples of a known base and multiplier with no need for known examples or prior knowledge of digits.

Paper Structure

This paper contains 7 sections, 7 theorems, 16 equations, 16 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Basic Notation, Definitions, and Results
  3. A Graph-Theoretical Approach
  4. A Method for Finding All Permutiples up to a Known Length
  5. Finite-State Machine Methods
  6. Application to Palintiple Numbers
  7. Summary, Conclusions, and Future Lines of Inquiry

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 (16)

  • 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 directed edges.
  • Figure 2: The $(2,6)$-mother graph.
  • Figure 3: An edge on the state diagram.
  • Figure 4: The $(2,4)$-Hoey-Sloane graph.
  • Figure 5: The $L$-walk of the $(2,4)$-permutiple string $(0,0)(2,3)(1,2)(3,1).$
  • ...and 11 more figures

Theorems & Definitions (28)

  • Definition 1: holt_3
  • Theorem 1: holt_3
  • Theorem 2: holt_3
  • Definition 2: holt_3
  • Definition 3
  • Theorem 3
  • Definition 4
  • Theorem 4
  • proof
  • Definition 5
  • ...and 18 more