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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.

##### MSC:
 11D61 Exponential Diophantine equations