Exact Polynomial Families Solving the Erdos-Straus Equation

Abstract

The Erd\H{o}s-Straus conjecture, proposed in 1948 by Paul Erd\H{o}s 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 $b, c, d \in \mathbb{N}^*$ for every integer $a \geq 2$. While the conjecture has been confirmed for all even integers and for all integers congruent to $3 \pmod{4}$, the case $a \equiv 1 \pmod{4}$ 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 $x, y, z \geq 1$, 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 Erd\H{o}s--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 \[ 4p_1(x,y,z)+1,\quad 4p_2(x,y,z)+1,\quad 4p_3(x,y,z)+1,\quad 4p_4(x,y,z)+1 \] collectively cover all integers congruent to $1 \pmod{4}$. 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.