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arXiv · 2606.10922

A Divisor Parametrization for the Erd\H{o}s--Straus Conjecture

M. Bello-Hernández·M. Benito·E. Fernández
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Abstract

We study representations of \(1/n\) as a sum of three unit fractions whose denominators are all divisible by a prescribed integer \(m\). After scaling, this is equivalent to representing \(m/n\) as a sum of three unit fractions. Our main focus is the Erd\H{o}s--Straus case \(m=4\). We introduce a divisor-based function \(\mathrm{fab}(n,a,b)\), prove that its admissible parameters recover exactly the decompositions of \(1/n\) with all three denominators divisible by \(4\), and compare this parametrization with well-known Type I/II descriptions.

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M. Bello-Hernández, M. Benito, E. Fernández. 2026-06-09. A Divisor Parametrization for the Erd\H{o}s--Straus Conjecture. https://arxiv.org/abs/2606.10922

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