A Multigraph Characterization of Permutiple Strings

TL;DR

This paper introduces a multigraph-based framework to characterize permutiple strings, providing new conditions for their existence beyond previous finite-state-machine approaches.

Contribution

It presents a novel multigraph characterization of permutiples, establishing sufficient conditions for their existence that extend prior finite-state-machine methods.

Findings

Provides a multigraph condition for permutiple existence

Extends previous finite-state-machine recognition methods

Offers a new theoretical framework for permutiple analysis

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.