On the Diophantine equation \(Cx^2+D=2y^q\)
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.
scientific article; zbMATH DE number 7345241
scientific article