A Study on Erd\H{o}s-Straus conjecture on Diophantine equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$

TL;DR

This paper investigates the Erd ext{o}s-Straus conjecture, aiming to prove it for all natural numbers by analyzing representations of 4/n as sums of three unit fractions, and re-examining Mordell's theorem with specific modular conditions.

Contribution

The study provides a new approach to verify the conjecture for large classes of numbers and discusses limitations in proving it for all cases up to 10^5.

Findings

The conjecture holds for many numbers except possibly certain primes of specific forms.

A new expression relating 4/n to sums involving divisibility conditions is proposed.

The conjecture cannot be proved for twelve specific values of l up to 10^5.

Abstract

The Erd\H{o}s-Straus conjecture is a renowned problem which describes that for every natural number $n~(\ge 2)$, $\frac{4}{n}$ can be represented as the sum of three unit fractions. The main purpose of this study is to show that the Erd\H{o}s-Straus conjecture is true. The study also re-demonstrates Mordell theorem which states that $\frac{4}{n}$ has a expression as the sum of three unit fractions for every number $n$ except possibly for those primes of the form $n\equiv r$ (mod 780) with $r=1^2,11^2,13^2,17^2,19^2,23^2$. For $l,r,a\in\mathbb{N}$; $\frac{4}{24l+1}-\frac{1}{6l+r}=\frac{4r-1}{(6l+r)(24l+1)}$ with $1\le r\le 12l$, if at least one of the sums in right side of the expression, say, $a+(4r-a-1),~1\le a\le 2r-1$ for at least one of the possible value of $r$ such that $a,(4r-a-1)$ divide $(6l+r)(24l+1)$; then the conjecture is valid for the corresponding $l$. However, in this way the conjecture can not be proved only twelve values of $l$ for $l$ up to $l=10^5$.