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

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 $$ |z|\leq\sqrt{(1+\sqrt{2})|ab|}.$$