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

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