On the general solutions of the diophantine equation \(ax^2+ 2bxy- kay^2= \pm 1\)
type: Item
instance of: scholarly article
title: On the general solutions of the diophantine equation \(ax^2+ 2bxy- kay^2= \pm 1\)
zbMATH Open document ID: 0961.11006
author: Lionel Bapoungué
published in: Expositiones Mathematicae
publication date: 2000-09-04T00:00:00Z
review text: Let \(a, b, k\) be positive integers, and \(k\) be one of the integers 2, 3, 7, 11, 19, 43, 67, 163; let \(\delta = ka^2 + b^2\). To the diophantine equation (1) \(ax^2 + 2bxy - kay^2 = \pm 1\) associate the Pellian equation (2) \(v^2 - \delta w^2 = -k\). The author studies the connection between these equations, and proves, for example, that, if \(\delta\) is nonsquare and (2) is solvable, then among pairs \((a, b)\) for which \(\delta = ka^2 + b^2\) with \(a\) odd, there is exactly one pair for which (1) is solvable. He shows how all solutions of (1) can be generated from a single one using the fundamental solution of \(p^2 - \delta q^2 = \pm 1\).
Mathematics Subject Classification ID: 11D09
zbMATH DE Number: 1455781
zbMATH Keywords: Pell equation
zbMATH Keywords: quadratic diophantine equation
zbMATH Keywords: fundamental solution
reviewed by: Edward J. Barbeau
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1455781