Structure and form of the solutions of the Erdos-Straus conjecture

TL;DR

This paper classifies specific solutions to the Erdos-Straus conjecture, focusing on solutions sharing structural properties and providing characterizations for difficult cases like p=1009.

Contribution

It introduces a classification of solutions based on their structure and characterizes difficult cases, expanding understanding of the conjecture's solution space.

Findings

Classified solutions of the form (du,dv,duv)

Characterized values of p like p=1009

Analyzed solutions with gcd(x,y,z)=x and variable coincidences

Abstract

In this paper we classify certain values of p that satisfy the Erdos-Straus conjecture, concerning the decomposition of fractions of the form 4/n as sum of three fractions with numerator identically equal to 1, not according to their modular similarity but to the fact that they share solutions with identical structure. We classify all solutions that satisfy that they are of the form (du,dv,duv) and find characterizations for values of p that were initially difficult to classify, such as p=1009. We also study the solutions (x,y,z) that satisfy that gcd(x,y,z)=x and, finally, we classify all the cases in which any of these two variables coincide with each other.