A Divisor Parametrization for the Erdős--Straus Conjecture

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 Erdős--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.