Partial Resolution of the Erdös-Straus, Sierpinski, and Generalized Erdös-Straus Conjectures Using New Analytical Formulas

TL;DR

The paper introduces two analytical formulas that partially resolve the Erdős-Straus and Sierpiński conjectures by reducing the problem to discriminant conditions and, in one case, a divisibility constraint. The first formula relies on a divisibility condition that yields explicit $x,y,z$ when $4x-n\, ext{±} abla$ divides $2nx$, while the second uses the existence of a perfect square in the discriminant to generate solutions via $y=t(4x-n)\!-q$ and $z=t(4x-n)+q$. These approaches extend to the generalized Erdős-Straus conjecture with a parameter $a\ge4$, and they also generalize to the Sierpiński case with $5/n$ in place of $4/n$, including special and general cases. The work provides substantial numerical evidence, including detailed tests against Mordell’s exceptional cases, and offers open questions about explicit constructions of the required parameters to achieve a full proof. Taken together, the methods offer a structured, algebraic route toward a complete resolution of these Egyptian-fraction conjectures and invite further theoretical and computational exploration.

Abstract

This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce two new analytical formulas under specific conditions of divisibility and the existence of perfect squares. Under these conditions, the formulas verify the conjectures even for very large numerical values. Moreover, our method reduces the problem to the search for a suitable perfect square, thereby opening the way to a complete proof of these conjectures. Notably, our second formula significantly improves upon Mordell's work by demonstrating analytically the conjecture in the majority of cases where Mordell's approach fails. Furthermore, these formulas are highly versatile, as they provide, under the established conditions, a systematic method to decompose any fraction a/n into a sum of three Egyptian fractions. In conclusion, we present open questions and conjectures to the mathematical community regarding the generalization of these formulas.