On the diophantine equation $x^2+2^a3^b73^c=y^n $

Murat Alan and Mustafa Aydin

Address: Yildiz Technical University, Department of Mathematics, Davutpasa Campus, 34210, Esenler, Istanbul, Turkey

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Abstract: 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.

AMSclassification: primary 11D61; secondary 11D59, 11Y50.

Keywords: diophantine equations, primitive divisor theorem, Ramanujan-Nagell equations.

DOI: 10.5817/AM2023-5-411