The Diophantine equation \(x^2+2^k=y^n\). II
type: Item
instance of: scholarly article
title: The Diophantine equation \(x^2+2^k=y^n\). II
zbMATH Open document ID: 0960.11025
DOI: 10.1155/S0161171299224593
author: J. H. E. Cohn
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2001-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/48711
review text: In the first paper in the series [\textit{J. H. E. Cohn}, Arch. Math. 59, 341-344 (1992; Zbl 0770.11019)], the author conjectured that the equation \[ x^2+ 2^{2m}= y^n, \quad x,y,m,n\in \mathbb{N}, \quad \text{gcd}(x,y)= 1,\;m> 2, \tag \(*\) \] has no solutions \((x,y,m,n)\). In this paper he proves that if \((x,y,m,n)\) is a solution of \((*)\), then either \(m= 3^{2\alpha-1} (24k+ 13)\) and \(n\equiv 127 \pmod {144}\) or \(m\equiv 0 \pmod {1020}\) and \(x\equiv 0\pmod {85}\), where \(\alpha\), \(k\) are positive integers. Reviewer's remark: The above-mentioned conjecture has been verified by the reviewer [On Cohn's conjecture concerning the Diophantine equation \(x^2+ 2^m= y^n\), Arch. Math. (to appear)]. This result implies that a related conjecture posed by \textit{S. A. Arif} and \textit{F. S. Abu Muriefah} [Int. J. Math. Math. Sci. 20, 299-304 (1997; Zbl 0881.11038)] is true.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1538433
zbMATH Keywords: exponential Diophantine equation
reviewed by: Le Maohua
MaRDI profile type: Publication
OpenAlex ID: W1579043960
description: scientific article; zbMATH DE number 1538433