Almost a Complete Proof of the Generalized Erdős-Straus Conjecture: $5\mathrm{/}a=1\mathrm{/}b+1\mathrm{/}c+1\mathrm{/}d$

The generalized Erdős-Straus conjecture, proposed by Wacław Sierpiński 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 $a\equiv i(\mathrm{m}\mathrm{o}\mathrm{d}5)$ for $i\in \{0,2,3,4\}$, providing direct decompositions. The remaining open case, $a\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$, is addressed for $a=5q+1$ with $q\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}252)$, where we give explicit decompositions, often with $q$ expressed as three-variable polynomials. For $q\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}252)$, we conjecture that a specific polynomial $p_1(x,y,z)=z(x(5y-1)-y)-x,x,y,z\in {\mathbb{N}}^{\ast }$, which exactly satisfies the generalized Erdős–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 $\mathit{\text{Mathematica}}$ implementation is included.