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Erdös–Straus conjecture (m = 4)

Prove that for every integer n ≥ 2 there exist integers x, y, and z such that 4/n = 1/x + 1/y + 1/z.

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Background

The paper studies polynomial solutions to the Diophantine equation m/n = 1/x + 1/y + 1/z in residue classes and discusses restrictions when m does not divide the modulus n1. As context, it recalls the classical Erdös–Straus conjecture for m = 4.

Despite extensive partial results and computational verifications reported in the literature, the conjecture has not been proven. The authors cite it to situate their results within the broader landscape of Egyptian fraction problems.

References

Two well known conjectures by Erdös-Straus and Sierpinski state that the diophantine equation m/ n =1/ x+1/ y+1/ z has integer solutions x, y,zfor every integer n ≥ 2and m = 4 (Erdös-Straus) or m = 5 (Sierpinski). There is an impressive body of evidence for the validity of both conjectures, see e.g. [1,2,3,5], but no valid proof.

— Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$ (2404.01307 - Schuh, 3 Mar 2024) in Section I (Introduction)