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On the number of solutions of the generalized Ramanujan-Nagell equation \(x^2+D=k^n\). (Chinese. English summary) Zbl 1005.11010

Summary: Let \(D\), \(k\) be positive integers with \(\gcd (D,k)= 1\). In this paper, using elementary methods, we prove that if \(D\) is a prime, then the equation \(x^2+ D= k^n\) has at most eight positive integer solutions \((x,n)\).

MSC:

11D61 Exponential Diophantine equations
11D45 Counting solutions of Diophantine equations
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