On the Diophantine equation \(Cx^2+D=2y^q\)
type: Item
instance of: scholarly article
title: On the Diophantine equation \(Cx^2+D=2y^q\)
zbMATH Open document ID: 1484.11099
author: Nejib Ghanmi
author: Fadwa S. Abu Muriefah
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: In the paper under review the authors study the Diophantine equation in the title where \(C,D\) are given integers and \(x,y,q\) are integer unknowns with \(q\ge 3\) prime. The main result of the authors is that if \(CD\) is positive and squarefree not congruent to \(3\pmod 4\) and \(q\) is coprime to the class number of the complex quadratic number field \({\mathbb Q}({\sqrt{-CD}})\), then \(q=3,5\). Assuming \(C,D\) positive and also \(x\) positive, if \(q=3\), then \(D=3a^2C+\varepsilon\) and \((x,y)=(a(4a^2C+3\varepsilon),2a^2C+\varepsilon)\) for some \(\varepsilon\in \{\pm 1\}\) and some positive integer \(a\), whereas if \(q=5\) then either \((D,C,x,y)=(11,7,1169,9)\) or \(2D=L_{k+\varepsilon}\) and \(2a^2C=F_{k-2\varepsilon}\) for some \(\varepsilon\in \{\pm 1\}\) and some positive integer \(a\), and \((x,y)=(a(2L_{2k+\varepsilon}+3\varepsilon (-1)^k)/5,F_k)\), where \(F_n,~L_n\) denote the \(n\)th Fibonacci and Lucas number, respectively. The proofs use the classification theorem of \textit{Yu. Bilu} et al. [J. Reine Angew. Math. 539, 75--122 (2001; Zbl 0995.11010)] of members of Lucas sequences with complex conjugated roots which have no primitive divisors.
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11Y50
zbMATH DE Number: 7345241
zbMATH Keywords: Diophantine equation
zbMATH Keywords: Fibonacci sequence
zbMATH Keywords: primitive divisor
zbMATH Keywords: Lehmer pair
zbMATH Keywords: Lucas sequence
reviewed by: Florian Luca
MaRDI profile type: Publication
full work available at URL: s11139-019-00165-w
OpenAlex ID: W2969363145
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DOI: 10.1007/S11139-019-00165-W
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description: scientific article; zbMATH DE number 7345241