Solutions in $\mathbb{Z}[i]$ of $A^{5}+B^{5}=C^{5}\pm 1$

TL;DR

This paper constructs solutions in Gaussian integers for the equation A^5 + B^5 = C^5 ± 1 using a method inspired by Ramanujan's identities, expanding understanding of fifth powers in complex integer rings.

Contribution

It introduces a novel application of Hirschhorn's method to find solutions in lgebraic integers for a specific Diophantine equation.

Findings

Constructed explicit solutions in lgebraic integers

Extended methods for solving fifth power equations in lgebraic number rings

Demonstrated the applicability of Ramanujan-inspired identities

Abstract

We build solutions in $\mathbb{Z}[i]$ of $A^{5}+B^{5}=C^{5}\pm 1$ based on a method developed by Michael D. Hirschhorn in his article "An Amazing Identity of Ramanujan", from Mathematics Magazine, Vol. 68, No3 (June 1995).