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

2014/05/08 by Jose Luis Leal Ruperto, Ruperto, Jose Luis Leal · 1 citation
Mathematics · #11D #11G #11Y #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1405.1949

openalex publication_date 2014/05/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Holzer proves that Legendre's equation ax²+by²+cz²=0, expressed in its normal form, when having a nontrivial solution in the integers, has a solution (x,y,z) where |x|≤√(|bc|), |y|≤√(|ac|), |z|≤√(|ab|). This paper proves a similar version of the theorem, for Legendre's equation with coefficients a, b,c in Gaussian integers ℤ[i] in which there is a solution (x,y,z) where |z|≤√((1+√(2))|ab|).

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