Egyptian fractions for few primes

TL;DR

This paper investigates Egyptian fraction solutions with denominators constrained to specific primes, assessing the greedy algorithm's effectiveness and introducing improved algorithms for generating solutions with prime restrictions.

Contribution

It provides new algorithms for generating low-rank Egyptian fraction solutions with prime constraints and evaluates their performance against existing methods.

Findings

Greedy algorithm's bounds are analyzed for prime-constrained Egyptian fractions.

New algorithms outperform previous methods in generating solutions.

Computational results demonstrate the efficiency of the proposed algorithms.

Abstract

We study solutions to the Egyptian fractions equation with the prime factors of the denominators constrained to lie in a fixed set of primes. We evaluate the effectiveness of the greedy algorithm in establishing bounds on such solutions. Additionally, we present improved algorithms for generating low-rank solutions and solutions restricted to specific prime sets. Computational results obtained using these algorithms are provided, alongside a discussion on their performance.