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On the Diophantine equation \(x^2+7^{2k}=y^n\). (English) Zbl 1221.11091
Here the authors prove that all solutions \((x,y,k)\) for \(n\geq 3\) to the Diophantine equation \[ x^2+7^{2k}=y^n \] in integers \(x\geq 1\), \(y\geq 1\), \(k\geq 1\) are given by \((524\cdot 7^{3\lambda},65\cdot 7^{2\lambda},1+3\lambda)\) for \(n=3\) and \((24\cdot 7^{2\lambda},5\cdot 7^ {\lambda},1+2\lambda)\) for \(n=4\) where \(\lambda\geq 0\) is any integer.

MSC:
11D61 Exponential Diophantine equations
11Y50 Computer solution of Diophantine equations
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