Research article
Special Issues
On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)
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Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang, Henan, 471023, China
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2.
Department of Mathematics and Computer, Hetao College, Bayannur, Inner Mongolia, 015000, China
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Received:
01 March 2021
Accepted:
19 July 2021
Published:
22 July 2021
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MSC :
11D61
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Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai [6] conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).
Citation: Yahui Yu, Jiayuan Hu. On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)[J]. AIMS Mathematics, 2021, 6(10): 10596-10601. doi: 10.3934/math.2021615
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Abstract
Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai [6] conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).
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