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On some generalized Lebesgue-Nagell equations. (English) Zbl 1219.11059
This paper continues the study of the Lebesgue-Nagell equation \(x^2+D=y^n\), where \(n\geq 3\). Here the Authors consider the special case \(D=q^m\) where \(q\) is a prime number \({}\geq 11\) and they solve completely this exponential Diophantine equation when the ring of integers of the quadratic field \({\mathbb Q}(\sqrt{-q})\) is principal, that is when \(q\in\{11,19,43,67,163\}\).
The proof combines deep (obtained by modular method and Baker’s theory) and elementary arguments.

MSC:
11D61 Exponential Diophantine equations
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