Abstract
Let a, b 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 (x, y, m). Moreover, under the assumption that b is an odd prime, there exist only finitely many triples (a, b, n) for which the equation has a solution (x, y, m) with \(m>1\).
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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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DOI: https://doi.org/10.1007/s10998-021-00419-5