On the diophantine equation \(x^ 2+2^ k=y^ n\)
type: Item
instance of: scholarly article
title: On the diophantine equation \(x^ 2+2^ k=y^ n\)
zbMATH Open document ID: 0881.11038
DOI: 10.1155/S0161171297000409
author: S. Akhtar Arif
author: Fadwa S. Abu Muriefah
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1997-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/47795
review text: In a recent paper \textit{J. H. E. Cohn} [Arch. Math. 59, 341-344 (1992; Zbl 0770.11019)] gave a complete solution of the equation \[ x^2+2^k= y^n, \] for \(k\) odd. He proved that if \(k\) is odd there are exactly three families of solutions. In the present paper the authors consider the case \(k\) even and \(m>1\). Their main result is that the equation of the title has two families of solutions given by \[ x=2^m, \qquad y^n=2^{2m+1} \] and by \[ m=3M+1, \quad n=3, \quad x=11\cdot 2^{3M}, \quad y=5\cdot 2^{2M} \] except possibly when \(m=3^{2k+1}m'\), \((m',3)=1\), \(x\) is odd and all prime divisors of \(n\) are congruent to \(7\pmod 8\). The proofs of all results are elementary and rely upon the factorisation of the title equation in the Gaussian field.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1046338
zbMATH Keywords: higher degree diophantine equations
zbMATH Keywords: Gaussian field
reviewed by: Ray Phillip Steiner
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1046338