On the Diophantine equation \(x^{2}+p^{2k+1}=4y^{n}\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \(x^{2}+p^{2k+1}=4y^{n}\)
zbMATH Open document ID: 1064.11029
DOI: 10.1155/S0161171202106107
author: S. Akhtar Arif
author: Amal S. Al-Ali
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-10-15T00:00:00Z
full work available at URL: https://eudml.org/doc/51345
review text: Summary: It has been proved that if \(p\) is an odd prime, \(y>1\), \(k\geq 0\), \(n\) is an integer greater than or equal to \(4\), \((n,3h)=1\) where \(h\) is the class number of the field \(\mathbb{Q}(\sqrt{-p})\), then the equation \(x^{2}+p^{ 2k+1}=4y^{n}\) has exactly five families of solution in the positive integers \(x\), \(y\). It is further proved that when \(n=3\) and \(p=3a^{2}\pm 4\), then it has a unique solution \(k=0\), \(y=a^{2}\pm 1\).
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1814808
MaRDI profile type: Publication
OpenAlex ID: W2015303439
description: scientific article; zbMATH DE number 1814808