Almost a Complete Proof of the Generalized Erd\H{o}s-Straus Conjecture: ${5}/{a} = {1}/{b} + {1}/{c} + {1}/{d}$

Abstract

The generalized Erd\H{o}s-Straus conjecture, proposed by Wac\l{}aw Sierpi\'{n}ski in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in \mathbb{N}$ for every integer $a \ge 2$. In this work we present explicit solutions for all integers $a \ge 2$. We begin with the simplest known cases where $a \equiv i \pmod{5}$ for $i \in \{0,2,3,4\}$, providing direct decompositions. The remaining open case, $a \equiv 1 \pmod{5}$, is addressed for $a = 5q + 1$ with $q \not\equiv 0 \pmod{252}$, where we give explicit decompositions, often with $q$ expressed as three-variable polynomials. For $q \equiv 0 \pmod{252}$, we conjecture that a specific polynomial $p_{1}(x,y,z)=z (x (5 y-1)-y)-x,~ x,y,z \in \mathbb{N}^*$, which exactly satisfies the generalized Erd\H{o}s--Straus equation, generates all such multiples of $252$. This conjecture has been verified computationally for $5q+1$ up to approximately $10^{10}$, and the corresponding \textit{Mathematica} implementation is included.