On a conjecture concerning the exponential Diophantine equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $

Shuanglin Fei; Guangyan Zhu; Rongjun Wu; Shuanglin Fei; Guangyan Zhu; Rongjun Wu

doi:10.3934/era.2024184

Electronic Research Archive

2024, Volume 32, Issue 6: 4096-4107. doi: 10.3934/era.2024184

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Research article

On a conjecture concerning the exponential Diophantine equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.
School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China
3.
School of Mathematics, Southwest Minzu University, Chengdu 610064, China

Received: 16 April 2024 Revised: 14 June 2024 Accepted: 20 June 2024 Published: 26 June 2024

Let $ a $, $ b $, $ c $, and $ n $ be positive integers such that $ a+b = c^{2} $, $ 2\nmid c $ and $ n > 1 $. In this paper, we prove that if $ \gcd(c, n) = 1 $ and $ n\geq 117.14c $, then the equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $ has only the positive integer solution $ (x, y, z) = (1, 1, 2) $ under the assumption $ \gcd(an^{2}+1, bn^{2}-1) = 1 $. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer $ n $, if $ (a, b, c) = (12, 13, 5) $, $ (18, 7, 5) $, or $ (44, 5, 7) $, then this equation has only the solution $ (x, y, z) = (1, 1, 2) $. This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.
- linear forms in $ m $-adic logarithms,
- exponential Diophantine equation,
- positive integer solution
Citation: Shuanglin Fei, Guangyan Zhu, Rongjun Wu. On a conjecture concerning the exponential Diophantine equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $[J]. Electronic Research Archive, 2024, 32(6): 4096-4107. doi: 10.3934/era.2024184

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Abstract

Let $ a $, $ b $, $ c $, and $ n $ be positive integers such that $ a+b = c^{2} $, $ 2\nmid c $ and $ n > 1 $. In this paper, we prove that if $ \gcd(c, n) = 1 $ and $ n\geq 117.14c $, then the equation $ (an^{2}+1)^{x}+(bn^{2}-1)^{y} = (cn)^{z} $ has only the positive integer solution $ (x, y, z) = (1, 1, 2) $ under the assumption $ \gcd(an^{2}+1, bn^{2}-1) = 1 $. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer $ n $, if $ (a, b, c) = (12, 13, 5) $, $ (18, 7, 5) $, or $ (44, 5, 7) $, then this equation has only the solution $ (x, y, z) = (1, 1, 2) $. This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.

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