On the magnitude of the gaussian integer solutions of the Legendre equation

TL;DR

This paper extends a classical result on bounds for solutions of Legendre's equation to the Gaussian integers, establishing a new upper bound for solutions in the complex integer domain.

Contribution

It provides a novel bound for solutions of Legendre's equation over Gaussian integers, generalizing Holzer's classical bounds in the integer case.

Findings

Established a bound |z| ≤ √( (1+√2)|ab| ) for solutions in Gaussian integers.

Generalized classical bounds from integers to Gaussian integers.

Contributed to the understanding of solutions in complex quadratic integer domains.

Abstract

Holzer proves that Legendre's equation $$ax^2+by^2+cz^2=0, $$ expressed in its normal form, when having a nontrivial solution in the integers, has a solution $(x,y,z)$ where $|x|\leq\sqrt{|bc|}, \quad |y|\leq\sqrt{|ac|}, \quad |z|\leq\sqrt{|ab|}.$ This paper proves a similar version of the theorem, for Legendre's equation with coefficients $a, b,c$ in Gaussian integers $\mathbb{Z}[i]$ in which there is a solution $(x,y,z)$ where $$ |z|\leq\sqrt{(1+\sqrt{2})|ab|}.$$