On the equation \(x^ 2+Dxy+y^ 2=z^ k\)
type: Item
instance of: scholarly article
title: On the equation \(x^ 2+Dxy+y^ 2=z^ k\)
zbMATH Open document ID: 0669.10036
author: Jerzy Wojtacha
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1988-01-01T00:00:00Z
review text: The author's result is the following. All integer solutions x,y,z of the equation \[ | x^ 2+Dxy+y^ 2| =z^ k \] where \(D=\pm 1,\pm 3,\pm 5,\pm 7,\pm 9\), \(k=1,2,3,...\), and \((x,y,z)=1\), \(3\nmid z\), are given by the formulae \[ | m^ 2+Dmn+n^ 2| =z,\quad \left( \begin{matrix} x\\ y\end{matrix} \begin{matrix} -y\\ x+Dy\end{matrix} \right)=\left( \begin{matrix} m\\ n\end{matrix} \begin{matrix} -n\\ m+Dn\end{matrix} \right)^ k\left( \begin{matrix} u\\ v\end{matrix} \begin{matrix} -v\\ u+Dv\end{matrix} \right), \] where m,n,u,v\(\in {\mathbb{Z}}\), \((m,n)=1\), \(| u^ 2+Duv+v^ 2| =1\). In the case \(D\neq \pm 7\), the assumption \(3\nmid z\) can be omitted.
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 4095573
zbMATH Keywords: matrix method
MaRDI profile type: Publication
Recommended article: Q4769093
Recommended article: On the equation x2 + ry2 = z2
Recommended article: Q4900055
Recommended article: Q5515474
Recommended article: On the equation \(y^x\pm x^y=z^2\)
Recommended article: On the equation \(x^ 4+mx^ 2y^ 2+y^ 4=z^ 2\)
Recommended article: ON THE EQUATION ax2+by2+cz2=0
Recommended article: On the equation \(x^ 2-Dy^ 2=z^ k\) with \(D=2,3,5,7,11,13\)
Recommended article: Q3124714
Recommended article: ON THE EQUATION x(x+d)…(x+(k-1)d)=by 2
reviewed by: Ke Zhao
description: scientific article; zbMATH DE number 4095573