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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
##### Keywords:
exponential Diophantine equations