Catalan's Equation For Even Primes over Quadratic Fields

TL;DR

This paper classifies solutions to Catalan's equation over quadratic integer rings when one prime exponent is even, expanding understanding of exponential Diophantine equations in algebraic number fields.

Contribution

It provides a complete list of solutions for Catalan's equation in Gaussian integers with one prime exponent even, a novel extension to previous results in quadratic fields.

Findings

All solutions for the equation with one even prime exponent are listed.

The solutions are explicitly characterized within Gaussian integers.

The work extends classical Catalan problem results to quadratic integer rings.

Abstract

This article lists all the solution of the Catalan equation $x^p-y^q=1$ for $x,y \in \mathbb{Z}[i]$, when one of the primes $p and q$ is even.