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

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...

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Bibliographic Details
Published inarXiv.org
Main Author Leal Ruperto, Jose Luis
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.05.2014
Subjects
Integers
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Summary: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 $$ |z|\leq\sqrt{(1+\sqrt{2})|ab|}.$$
ISSN:2331-8422