Small solutions of the Legendre equation
type: Item
instance of: scholarly article
title: Small solutions of the Legendre equation
zbMATH Open document ID: 0908.11012
author: Todd Cochrane
author: Patrick Mitchell
published in: Journal of Number Theory
publication date: 1999-03-18T00:00:00Z
review text: Let \(a\), \(b\), \(c\) be positive integers. The Legendre equation \(ax^2+ by^2- cz^2= 0\) is said to be in normal form if \(a\), \(b\), \(c\) are squarefree and pairwise coprime. If the Legendre equation \(ax^2+ by^2- cz^2= 0\) is in normal form and has a solution in integers \(x\), \(y\), \(z\) not all zero, then there is a nontrivial solution with \(| x|\leq \sqrt{bc}\), \(| y|\leq \sqrt{ac}\) and \(| z|\leq \sqrt{ab}\). This result was first proved by \textit{L. Holzer} [Can. J. Math. 2, 238-244 (1950; Zbl 0037.02602)] using a deep result. The first elementary proof was given by \textit{L. J. Mordell} [J. Number Theory 1, 1-3 (1969; Zbl 0169.06104)] with a small gap filled by \textit{K. S. Williams} [Util. Math. 34, 65-72 (1988; Zbl 0666.10010)]. In this paper, the authors give a new elementary proof of this theorem by showing that the lattice of solutions of the congruence \[ ax^2+ by^2- cz^2\equiv 0\pmod{abc}, \] which is constructed in most textbook proofs of Legendre's necessary and sufficient condition for the solvability of \(ax^2+ by^2- cz^2\equiv 0\), always contains a solution satisfying the above bounds.
reviewed by: Kenneth S. Williams
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 11D79
zbMATH DE Number: 1207965
zbMATH Keywords: small solutions
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: congruences
zbMATH Keywords: Legendre equation
MaRDI profile type: Publication
full work available at URL: jnth.1998.2221
OpenAlex ID: W1975287686
cites work: An Introduction to the Geometry of Numbers
cites work: Minimal Solutions of Diophantine Equations
cites work: On the magnitude of the integer solutions of the equation \(ax^ 2 +by^ 2 +cz^ 2 = 0\)
cites work: Q3818387
DOI: 10.1006/JNTH.1998.2221
description: scientific article; zbMATH DE number 1207965