# Digit Preserving Multiplication in Continued Fraction Representations

## TL;DR

This paper explores when continued fraction representations of rational numbers are multiples of permutations of their partial quotients, developing classifications and generating new examples through mathematical analysis.

## Contribution

It introduces the concept of permutiples in continued fractions, classifies 2-4 digit cases, and provides methods to generate new permutiples from existing ones.

## Key findings

- Classified all 2, 3, and 4-digit continued fraction permutiples.
- Developed conditions for generating new permutiples via digit concatenation.
- Provided examples and general results for permutiple structures.

## Abstract

A permutiple is a number which is an integer multiple of some permutation of its digits. A well-known example is 9801 since it is an integer multiple of its reversal, 1089. In this paper, we consider the permutiple problem in an entirely different setting: continued fractions. We pose the question of when the simple continued fraction representation of a rational number is an integer multiple of a permutation of its partial quotients (or digits, as we shall call them). We develop some general results and apply them to finding new examples. In doing so, we attempt to classify all 2, 3, and 4-digit continued fraction permutiples in terms of basic permutiple types which we discover along the way. We also generate new examples from old by finding conditions which guarantee that digit-string concatenation yields other permutiples.