On Permutiples Having a Fixed Set of Digits

TL;DR

This paper investigates the properties of permutiples, numbers that are multiples of permutations of their digits, and develops methods to find new examples with the same digit set by analyzing their digits and carries.

Contribution

It introduces a general framework for digit preserving multiplication and provides methods to generate new permutiples from known examples with identical digit sets.

Findings

Digits and carries of permutiples are related systematically.

Methods for deriving new permutiples from existing ones are developed.

Focus on permutiples with a fixed set of digits.

Abstract

A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently, palintiple numbers, have been well-studied. In this paper we study the problem of general digit preserving multiplication. We show how the digits and carries of a permutiple are related and utilize these relationships to develop methods for finding new permutiple examples from old. In particular, we shall focus on the problem of finding new permutiples from a known example having the same set of digits.