Exact Polynomial Families Solving the Erdos-Straus Equation

The Erdős-Straus conjecture, proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the Diophantine equation $$\frac{4}{a}=\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$$ admits positive integer solutions $b,c,d\in {\mathbb{N}}^{\ast }$ for every integer $a\ge 2$. While the conjecture has been confirmed for all even integers and for all integers congruent to $3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the case $a\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ remains the central open challenge. In this work, we construct four explicit unbounded multivariable polynomials $p_1(x,y,z),p_2(x,y,z),p_3(x,y,z),p_4(x,y,z)$ with $x,y,z\ge 1$, such that each of the first three – when inserted into the form $a=4p_i(x,y,z)+1$ – always produces values of $a$ for which the Erdős–Straus equation admits an explicit solution. Thus, the first three polynomials individually satisfy the conjecture for all their outputs. We further conjecture that the values $$4p_1(x,y,z)+1,4p_2(x,y,z)+1,4p_3(x,y,z)+1,4p_4(x,y,z)+1$$ collectively cover all integers congruent to $1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Extensive computational verification up to $q={10}^9$ confirms that every integer of the form $4q+1$ within this range arises from at least one of these families. One of the polynomials alone generates all such prime values up to at least $1.2\times {10}^{10}$. These results offer strong computational evidence and explicit constructions relevant to the resolution of the conjecture.