A unified parametric approach to the Erd\H{o}s--Straus conjecture with explicit solutions for a set of integers of natural density one

TL;DR

This paper introduces a parametric method to analyze the Erdős–Straus conjecture, providing explicit solutions for most integers and verifying the conjecture for a set of natural density one.

Contribution

It develops a unified parametric framework and constructs explicit solutions for a large class of integers, advancing understanding of the conjecture's validity.

Findings

Explicit solutions for 75% of integers in certain residue classes

Verification of the conjecture for a set of integers with density one

Introduction of a fundamental function whose perfect square condition yields solutions

Abstract

We develop a parametric approach to study the Diophantine equation $\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$, underlying the Erd\H{o}s--Straus ($k=4$), Sierpi\'nski ($k=5$), and related generalizations. We introduce and analyze the properties of the fundamental function $F_{x,t}^{(k)}(n) = t^2(kx-n)^2 - 2nxt$, whose being a perfect square is equivalent to yielding a solution of these conjectures. In the classical Erd\H{o}s--Straus case ($k=4$), for the residue classes $n \equiv 0,2,3 \pmod{4}$, we provide explicit symmetric solutions $y=z$, covering already 75\% of all integers. For the historically most resistant class $n \equiv 1 \pmod{4}$, we construct explicit symmetric solutions based on the existence of a divisor $b \equiv 3 \pmod{4}$, and we further show that this condition is satisfied for almost all such integers: the set of exceptions has natural density zero. Consequently, the Erd\H{o}s--Straus conjecture is verified for a proportion of integers tending to $1$ in this class. These results yield infinitely many new families of explicit solutions not covered by previous constructions, highlight the structural behavior of $F$.