A unified parametric approach to the Erdős--Straus conjecture with explicit solutions for a set of integers of natural density one

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 Erdős--Straus ($k=4$), Sierpiński ($k=5$), 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 Erdős--Straus case ($k=4$), for the residue classes $n\equiv 0,2,3(\mathrm{mod}4)$, we provide explicit symmetric solutions $y=z$, covering already 75\% of all integers. For the historically most resistant class $n\equiv 1(\mathrm{mod}4)$, we construct explicit symmetric solutions based on the existence of a divisor $b\equiv 3(\mathrm{mod}4)$, and we further show that this condition is satisfied for almost all such integers: the set of exceptions has natural density zero. Consequently, the Erdős--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 $F$.