Abstract
Let c be a fixed positive integer with \(c>1\). Very recently, Terai et al. (Int Math Forum 17:1–10, 2022) conjectured that the equation \(x^2+(4c)^y=(c+1)^z\) has only one positive integer solution \((x,y,z)=(c-1,1,2)\), except for \(c \in \{5,7,309\}\). In this paper, combining certain known results on Diophantine equations with some elementary methods, we verify that this conjecture is true for several cases.
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Acknowledgements
The authors thank the referee for helpful suggestions on the proof of Lemma 2.3, which made the proof consice.
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Communicated by B. Sury.
Nobuhiro Terai is supported by JSPS KAKENHI Grant Number 22K03271.
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Fujita, Y., Le, M. & Terai, N. A note on the solution to the generalized Ramanujan–Nagell equation \(\pmb {x^2+(4c)^y=(c+1)^z}\). Indian J Pure Appl Math 54, 1145–1157 (2023). https://doi.org/10.1007/s13226-022-00328-4
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DOI: https://doi.org/10.1007/s13226-022-00328-4