Well-defined Erd\"{o}s-Straus equations and L.C.M

TL;DR

This paper explores the Erdős-Straus conjecture by generalizing the equations involved, analyzing well-defined forms, and demonstrating the limitations of current proof methods, linking the problem to concepts like LCM and GCD.

Contribution

It introduces generalized well-defined equations related to the Erdős-Straus conjecture and shows the limitations of existing proof techniques, proposing related conjectures and focusing on LCM and GCD.

Findings

Certain solution methods are inefficient for proving the conjecture.

The conjecture cannot be fully proven using current methods.

Well-defined equations relate closely to LCM and GCD concepts.

Abstract

The Erd\"{o}s-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erd\"{o}s-Straus equation, state several methods for obtaining well-defined equations, and some features of well-defined equations. Using well-defined equations, we explain the inefficiency of the solutions that did not lead to a complete proof of the problem. Finally, we prove that this conjecture cannot be completely proven by the methods expressed in various articles. For this reason, we express conjectures equivalent to the Erd\"{o}s-Straus conjecture and make the concept of conjecture clearer. The well-defined equations of Erd\"{o}s-Straus have a lot to do with the concept of the least common multiple and the greatest common divisor, and in the last section, we will independently express the functions and features based on the subject of the least common multiple.