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A note on the Lebesgue–Ljunggren–Nagell equation \(ax^2+b^{2m}=4y^n\)

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Abstract

Let ab be fixed positive integers such that \(2\not \mid {ab}\), \(\gcd (a,b)=1\) and a is square-free, and let \(h(-a)\) denote the class number of the imaginary quadratic field \({\mathbb {Q}} (\sqrt{-a})\). Further, let n be a fixed prime satisfying \( n>3\), \(n\not \mid h(-a)\) and \(n\not \mid b\). In this paper, we investigate the solvability of the equation \(ax^2+b^{2m}=4y^n\), \(x,y,m\in {\mathbb {N}}\), \(\gcd (x,y)=1\). We prove that if every prime divisor p of b satisfies \(p\not \equiv \pm 1 \pmod {n}\), then the equation has no solutions (xym). Moreover, under the assumption that b is an odd prime, there exist only finitely many triples (abn) for which the equation has a solution (xym) with \(m>1\).

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Acknowledgements

Here, I would like to thank the reviewers for their suggestions.

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Correspondence to Deli Lei.

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This work is supported by P. N. S. F. (2019JM573), P. N. S. F. F. (370012000002) and X.M. U. F. (2018XNRC05)

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Guo, X., Lei, D. A note on the Lebesgue–Ljunggren–Nagell equation \(ax^2+b^{2m}=4y^n\). Period Math Hung 85, 72–80 (2022). https://doi.org/10.1007/s10998-021-00419-5

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