The Diophantine equation $x^4\pm y^4=iz^2$ in Gaussian integers

TL;DR

This paper completely solves the Diophantine equation $x^4\u00b1 y^4=iz^2$ over Gaussian integers using elliptic curves, and provides a new proof for the classical result on $x^4\u00b1 y^4=z^2$ solutions.

Contribution

It introduces a novel elliptic curve approach over b9; providing explicit solutions and a new proof for classical equations in Gaussian integers.

Findings

All solutions to $x^4\u00b1 y^4=iz^2$ are characterized.

The method offers a new proof for the triviality of solutions to $x^4\u00b1 y^4=z^2$ in Gaussian integers.

Elliptic curves over b9; are effectively used to analyze these Diophantine equations.

Abstract

In this note we find all the solutions of the Diophantine equation $x^4\pm y^4=iz^2$ using elliptic curves over $\mathbb Q(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\pm y^4=z^2$ has only trivial solutions in Gaussian integers.