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Keywords:
diophantine equations; primitive divisor theorem; Ramanujan-Nagell equations
diophantine equations; primitive divisor theorem; Ramanujan-Nagell equations
Summary:
In this paper, we find all integer solutions $ (x, y, n, a, b, c) $ of the equation in the title for non-negative integers $ a, b$ and $ c $ under the condition that the integers $ x $ and $ y $ are relatively prime and $ n \ge 3$. The proof depends on the famous primitive divisor theorem due to Bilu, Hanrot and Voutier and the computational techniques on some elliptic curves.
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