Solution of certain Diophantine equations in Gaussian integers

TL;DR

This paper investigates specific quartic Diophantine equations over Gaussian integers, demonstrating they have only trivial solutions under certain prime conditions by analyzing associated elliptic curves.

Contribution

It introduces a novel approach using elliptic curve torsion and rank calculations over Gaussian fields to solve particular Diophantine equations.

Findings

Only trivial solutions exist for the studied equations under given prime conditions.

Determined the torsion groups of related elliptic curves.

Calculated the rank of elliptic curves over (i).

Abstract

In this article, we show that the quartic Diophantine equations $x^4 \pm pqy^4=\pm z^2$ and $ x^4 \pm pq y^4= \pm iz^2$ have only trivial solutions for some primes $p$ and $q$ satisfying conditions $ p \equiv 3 \pmod 8, ~ q \equiv 1 \pmod 8 ~\text{and}~ \displaystyle\legendre{p}{q} = -1$. Here we have found the torsion of the two families of elliptic curves to find the solutions of given Diophantine equations. Moreover, we also calculate the rank of these two families of elliptic curves over the Gaussian field $\mathbb{Q}(i)$.