Exact Polynomial Families Solving the Erdos-Straus Equation

TL;DR

This paper constructs explicit polynomial families that generate infinite classes of integers for which the Erdős–Straus conjecture holds, providing computational evidence and new approaches towards solving the conjecture.

Contribution

The authors introduce four explicit unbounded polynomial families that produce integers satisfying the Erdős–Straus conjecture, covering all relevant cases up to very large bounds.

Findings

Polynomials generate all integers of the form 4q+1 up to 10^9.

One polynomial generates all such primes up to 1.2×10^{10}.

Computational evidence supports the conjecture's validity for these families.

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.