Representations of integers as quotients of sums of distinct powers of three

TL;DR

This paper explores which integers can be expressed as quotients of sums of distinct powers of three, providing initial conditions, an algorithm for representation, and a classification of such representations for integers up to 364.

Contribution

It introduces a necessary condition, develops an algorithm to determine and generate all representations, and classifies these representations based on their polynomial connections.

Findings

Characterization of representable integers up to 364

Algorithm to determine and generate representations

Classification of representations based on polynomial connections

Abstract

Which integers can be written as a quotient of sums of distinct powers of three? We outline our first steps toward an answer to this question, beginning with a necessary and almost sufficient condition. Then we discuss an algorithm that indicates whether it is possible to represent a given integer as a quotient of sums of distinct powers of three. When the given integer is representable, this same algorithm generates all possible representations. We develop a categorization of representations based on their connections to $0,1$-polynomials and give a complete description of the types of representations for all integers up to 364. Finally, we discuss in detail the representations of 7, 22, 34, 64, and 100, as well as some infinite families of integers.