Solutions of some generalized Ramanujan-Nagell equations via binary quadratic forms. (English) Zbl 1164.11020

Let \(d\) be an odd integer, and let \(h\) denote the class number of binary quadratic forms of discriminant \(-4d\). Write \(I\) for the identity form \(x^2+dy^2\). Suppose that \(\lambda\) is a prime power and \(k\) is a prime such that both \(\lambda\) and \(\lambda k^n\) are represented by \(I\). The authors show that there exists an integer \(r\) dividing \(h\) such that \(k^r\) is represented by \(I\), furthermore, the representations of \(\lambda k^n\) by \(I\) arise from the representations of \(\lambda\) and \(k^r\) by \(I\). As an important application, several generalized Ramanujan-Nagell equations of the form \(x^2+d=\lambda k^n\) are solved.


11D61 Exponential Diophantine equations