Rational numbers with odd greedy expansion of fixed length

TL;DR

This paper investigates the properties of odd greedy expansions of positive rational numbers with odd denominators, specifically characterizing those with fixed-length expansions and their initial denominators.

Contribution

It provides a complete classification of rational numbers with fixed-length odd greedy expansions and explores the structure of such expansions based on initial denominators.

Findings

Characterization of fractions with length 2 odd greedy expansion

Methods to find all fractions with specified initial denominators for fixed length

Existence of infinite families of fractions with prescribed initial denominators and fixed expansion length

Abstract

Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as long as the sum is less than $n/d$. It is an open question whether this expansion always has finitely many terms. Given a fixed positive integer $n$, we find all reduced fractions with numerator $n$ whose odd greedy expansion has length $2$. Given $m-1$ odd positive integers, we find all rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators. Given $m-2$ compatible odd positive integers, we find an infinite family of rational numbers whose odd greedy expansion has length $m$ and begins with these numbers as denominators.