On the number of solutions of the generalized Ramanujan-Nagell equation (Q2744709)

Let \(D_1,D_2,k\) be coprime positive integers, \(k\geq 2\), and let \(\lambda=2\) if \(k\) is even and \(\lambda=1\) or \(\sqrt{2}\) if \(k\) is odd. The authors study the number \({\mathcal N} (\lambda, D_1,D_2,k)\) of positive integer solutions \((x,n)\) of the equation (1) \(D_1x^2+D_2= \lambda^2k^n\). NEWLINENEWLINENEWLINEIn a series of papers, Le Maohua has proved that, for a prime \(p\), \({\mathcal N}(\lambda, D_1,D_2,p)\leq 2\) except for the cases \({\mathcal N}(2,1,7,2)= 5\) and \({\mathcal N}(2,3,5,2)= {\mathcal N}(2,1,11,3)= {\mathcal N}(2,1,19,5)= 3\). This result is sharp, in the sense that there are infinitely many \((D_1,D_2,k)\), such that \({\mathcal N}(\lambda, D_1,D_2,k)= 2\). On the other hand, by previous work of Bender and Herzberg and of Le Maohua, if either \(p\) is sufficiently large compared to \(D_1D_2\) or \(\max\{D_1,D_2,p\}\) is sufficiently large, then \({\mathcal N}(1, D_1,D_2,p)\leq 1\), unless \((D_1,D_2,p)\) belong to certain infinite families of values \((D_1,D_2,k)\). In this paper the authors succeed to determine explicitly all cases with \({\mathcal N}(\lambda, D_1,D_2,p)\geq 1\). NEWLINENEWLINENEWLINEThe detailed description of these cases, though too technical to be included in a short review, is thoroughly explicit. The authors adopt the classical approach to the problem via the theory of binary quadratic forms introduced by Bender and Herzberg in 1979. At some places they have to find all integer solutions of equations of the form \(f(x)= cy^n\), where \(c=1,2\) or 4 and \(f(x)\) is a quadratic polynomial. They do this based on previous work by other authors on such equations. At other places they are heavily based on a recent very strong result of \textit{Y. Bilu, G. Hanrot} and \textit{P. Voutier} [J. Reine Angew. Math. 539, 75-122 (2001; Zbl 0995.11010)], which is an indispensable tool for the proof of the main result of the present paper. NEWLINENEWLINENEWLINESeven interesting corollaries are given, for example: If \(\omega(k)\) denotes the number of distinct prime divisors of \(k\), then it is proved that equation (1) has at most \(2^{\omega(k)-1}\) solutions, with the exception of three infinite families of equations, which are explicitly described. Also, the only solutions of the equation \(x^2+7= 4y^n\) in integers \(x\geq 1\), \(y\geq 2\), \(n\geq 2\) are \((x,y,n)= (3,2,2), (5,2,3), (11,2,5), (181,2,13)\).