Integral solutions of Markoff-Hurwitz equations
type: Item
instance of: scholarly article
title: Integral solutions of Markoff-Hurwitz equations
zbMATH Open document ID: 0820.11016
author: Arthur Baragar
published in: Journal of Number Theory
publication date: 1995-09-14T00:00:00Z
review text: The author discusses the positive integral solutions to the symmetric equations of Hurwitz, \(x_ 1^ 2+ \dots+ x_ n^ 2= ax_ 1\cdots x_ n\). Markoff showed that in the case of \(a=n =3\), there is a fundamental solution from which a straightforward group action generates all of the positive integral solutions. Hurwitz proved that the set of positive integral solutions for a general pair \(a\) and \(n\) is either empty, or arises from a finite number of fundamental solutions. In this well-written paper, the author sharpens results of N. P. Herzberg on the set of \(a\) and \(n\) which lead to non-trivial solution sets. He also constructs, in two rather different manners (one completely elementary, the other involving the theory of quadratic number fields), sequences of \(a\) and \(n\) with unbounded number of fundamental solutions.
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 653305
zbMATH Keywords: quadratic diophantine equations
zbMATH Keywords: Markoff numbers
zbMATH Keywords: positive integral solutions
zbMATH Keywords: symmetric equations of Hurwitz
zbMATH Keywords: fundamental solutions
reviewed by: Thomas A. Schmidt
MaRDI profile type: Publication
full work available at URL: jnth.1994.1078
OpenAlex ID: W1988983085
DOI: 10.1006/JNTH.1994.1078
description: scientific article; zbMATH DE number 653305