Abstract
Let \(a,\ b,\ c,\ m\) be positive integers such that \(a+b=c^2, 2\mid a, 2\not \mid c\) and \(m>1\). In this paper we prove that if \(c\mid m \) and \(m>36c^3 \log c\), then the equation \((am^2+1)^x+(bm^2-1)^y=(cm)^z\) has only the positive integer solution \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).
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Acknowledgements
The authors would like to thank the referees for their valuable suggestions. This work is supported by N.S.F.(11226038, 11371012) of P.R. China , the Education Department Foundation of Shaanxi Province(14JK1311) and Scientific Research Foundation for Doctor of Xi’an Shiyou University (2015BS06)
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Fu, R., Yang, H. On the exponential diophantine equation \(\left( am^{2}+1\right) ^{x}+\left( bm^{2}-1\right) ^{y}=(cm)^{z}\) with \( c\mid m \) . Period Math Hung 75, 143–149 (2017). https://doi.org/10.1007/s10998-016-0170-z
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DOI: https://doi.org/10.1007/s10998-016-0170-z
Keywords
- Exponential diophantine equation
- Existence of primitive divisor of Lucas and Lehmer numbers
- Application of BHV theorem