Solutions to $\sum_{i=1}^n 1/x_i=1$ in integers $p^a\,q^b$ with $p$ and $q$ two set primes

TL;DR

This paper introduces an algorithm to find all integer solutions to the equation summing unit fractions with denominators as products of two distinct primes, focusing on solutions involving prime powers.

Contribution

The paper presents a novel algorithm for systematically computing solutions to a specific class of Egyptian fraction equations involving prime power denominators.

Findings

Algorithm efficiently finds all solutions with denominators as prime powers

Complete enumeration of solutions for given prime pairs

Provides a new method for solving prime-based Egyptian fractions

Abstract

We present an algorithm for computing all the solutions in not necessarily distinct integers to the decomposition of the unit into a sum of unit fractions with denominators $p^a.q^b$ where $p$ and $q$ are two distinct primes, each appearing at least once in the solution.