Gaussian integer solutions for the fifth power taxicab number problem

TL;DR

This paper explores solutions to the fifth power taxicab problem within Gaussian integers, expanding the scope of known solutions and introducing new methods for finding such solutions in complex integer domains.

Contribution

It introduces novel solutions to the fifth power taxicab problem involving Gaussian integers, extending the classical problem into complex integer settings.

Findings

Solutions found where $a$, $b$$ are positive integers and $c$, $d$ are Gaussian integers.

Solutions where all of $a$, $b$, $c$, and $d$ are Gaussian integers.

New methods for identifying solutions in complex integer domains.

Abstract

The famous open problem of finding positive integer solutions to $a^5 + b^5 = c^5 + d^5$ is considered, and related solutions are found in two distinct settings: firstly, where $a$ and $b$ are both positive integers with $c$ and $d$ both Gaussian integers; secondly, where all of $a$, $b$, $c$, and $d$ are Gaussian integers.