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Sierpinski conjecture (m = 5)

Prove that for every integer n ≥ 2 there exist integers x, y, and z such that 5/n = 1/x + 1/y + 1/z.

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Background

The authors introduce two classical problems on expressing rational numbers as sums of three unit fractions, including Sierpinski’s conjecture for m = 5, while developing results on polynomial parameterizations for general m ≥ 4 under certain modular constraints.

They note that, although many instances are known, a general proof of the conjecture remains elusive, providing motivation for their structural results on polynomial solutions.

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)