Unit Fractions in Norm-Euclidean Rings of Integers

TL;DR

This paper extends the Erdős-Straus conjecture to Gaussian integers and solves the related Diophantine equation over norm-Euclidean quadratic integer rings, advancing understanding of unit fractions in these algebraic structures.

Contribution

It introduces a Gaussian integer analogue of the Erdős-Straus conjecture and provides solutions over norm-Euclidean quadratic integer rings, a novel extension of the classical problem.

Findings

Established a Gaussian integer version of the Erdős-Straus conjecture.

Solved the Diophantine equation in norm-Euclidean quadratic rings.

Extended the understanding of unit fractions in algebraic number fields.

Abstract

In this paper we make a Gaussian integer version of the Erd\H{o}s-Straus conjecture and we solve the Erd\H{o}s-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields.