An algorithm for Egyptian fraction representations with restricted denominators

TL;DR

This paper introduces a Scheme-based algorithm that efficiently finds all Egyptian fraction representations of a rational number using a specified set of denominators, especially useful for dense representations with maximum denominator constraints.

Contribution

It presents a novel algorithm capable of generating all such representations with restricted denominators, addressing limitations of previous methods.

Findings

Algorithm finds all representations with given denominators

Efficiently handles dense representations with maximum denominator

Implemented in Scheme for accessibility

Abstract

A unit fraction representation of a rational number $r$ is a finite sum of reciprocals of positive integers that equals $r$. Of particular interest is the case when all denominators in the representation are distinct, resulting in an Egyptian fraction representation of $r$. Common algorithms for computing Egyptian fraction representations of a given rational number tend to result in extremely large denominators and cannot be adapted to restrictions on the allowed denominators. We describe an algorithm for finding all unit fraction representations of a given rational number using denominators from a given finite multiset of positive integers. The freely available algorithm, implemented in Scheme, is particularly well suited to computing dense Egyptian fraction representations, where the allowed denominators have a prescribed maximum.