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

Jose Luis Leal Ruperto

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arXiv · 1405.1949

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

Jose Luis Leal Ruperto

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

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Jose Luis Leal Ruperto. 2014-05-08. On the magnitude of the gaussian integer solutions of the Legendre equation. https://arxiv.org/abs/1405.1949

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