Partial Resolution of the Erd\"os-Straus, Sierpinski, and Generalized Erd\"os-Straus Conjectures Using New Analytical Formulas

TL;DR

This paper introduces new analytical formulas and a unified approach that partially resolve the Erd"os-Straus and Sierpinski conjectures, reducing the problem to finding suitable perfect squares and suggesting a path toward a full proof.

Contribution

It presents an equivalent reformulation of the conjectures and introduces two explicit formulas based on divisibility and perfect squares, offering a novel analytical framework.

Findings

Formulas verify the conjectures for large values

Reduction of the problem to finding a perfect square

Potential pathway to a complete proof

Abstract

This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two new explicit analytical formulas. The first formula, which is a special case of the second, is based on a divisibility condition, whereas the second, more general formula, relies on the existence of a perfect square, which we conjecture to always hold. 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. In conclusion, we present open questions and conjectures to the mathematical community regarding the generalization of these formulas.