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A Method for Finding All Permutiples with a Fixed Set of Digits from a Single Known Example

Benjamin V. Holt

TL;DR

The paper addresses the problem of enumerating all permutiples that share a fixed multiset of digits, multiplier, and length, starting from a single known example. It introduces a carry-based, permutation-centered framework that links digits, carries, and base permutations through linear relations in permutation matrices, notably $(bP_\psi-I)\mathbf{c}=(nP_\sigma-I)\mathbf{d}$ and $(nP_\tau-I)P_\pi\mathbf{d}=(bP_\psi-I)\hat{\mathbf{c}}$, and uses modular reduction to constrain base permutations. By computing possible base permutations and corresponding carry vectors, the method recovers all feasible $\pi$ and $\tau$, thereby producing all permutiples with the same digits, multiplier, and length as the known example. The approach is demonstrated on a base-10 example, recovering all $(4,10,\tau)$-permutiples with digits $\{8,7,9,1,2\}$, and on a base-13 example with repeated digits, revealing additional conjugacy classes. This constitutes a general, constructive tool for systematically enumerating permutiples beyond previous partial results, with potential connections to conjugacy structure and graph-theoretic models like Young graphs.

Abstract

A permutiple is a natural number that is a nontrivial multiple of a permutation of its digits in some base. Special cases of permutiples include cyclic numbers (multiples of cyclic permutations of their digits) and palintiple numbers (multiples of their digit reversals). While cyclic numbers have a fairly straightforward description, palintiple numbers admit many varieties and cases. A previous paper attempts to get a better handle on the general case by constructing new examples of permutiples with the same set of digits, multiplier, and length as a known example. However, the results are not sufficient for finding all possible examples except when the multiplier divides the base. Using an approach based on the methods of this previous paper, we develop a new method which enables us to find all examples under any conditions.

A Method for Finding All Permutiples with a Fixed Set of Digits from a Single Known Example

TL;DR

The paper addresses the problem of enumerating all permutiples that share a fixed multiset of digits, multiplier, and length, starting from a single known example. It introduces a carry-based, permutation-centered framework that links digits, carries, and base permutations through linear relations in permutation matrices, notably and , and uses modular reduction to constrain base permutations. By computing possible base permutations and corresponding carry vectors, the method recovers all feasible and , thereby producing all permutiples with the same digits, multiplier, and length as the known example. The approach is demonstrated on a base-10 example, recovering all -permutiples with digits , and on a base-13 example with repeated digits, revealing additional conjugacy classes. This constitutes a general, constructive tool for systematically enumerating permutiples beyond previous partial results, with potential connections to conjugacy structure and graph-theoretic models like Young graphs.

Abstract

A permutiple is a natural number that is a nontrivial multiple of a permutation of its digits in some base. Special cases of permutiples include cyclic numbers (multiples of cyclic permutations of their digits) and palintiple numbers (multiples of their digit reversals). While cyclic numbers have a fairly straightforward description, palintiple numbers admit many varieties and cases. A previous paper attempts to get a better handle on the general case by constructing new examples of permutiples with the same set of digits, multiplier, and length as a known example. However, the results are not sufficient for finding all possible examples except when the multiplier divides the base. Using an approach based on the methods of this previous paper, we develop a new method which enables us to find all examples under any conditions.
Paper Structure (4 sections, 3 theorems, 50 equations)

This paper contains 4 sections, 3 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Basic Notation, Definitions, and Results
  3. A Method for Finding All Examples
  4. Summary of Method and Concluding Remarks

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 $0\leq j \leq k$.

Theorems & Definitions (10)

  • Definition 1: holt_3
  • Theorem 1: holt_3
  • Theorem 2: holt_3
  • Definition 2: holt_3
  • Theorem 3
  • Example 1
  • Remark 1
  • Remark 2
  • Example 2
  • Remark 3