Abstract
Using the theory of Pellian equations, we show that the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) have infinitely many nontrivial integer solutions (x, y, z) for three classes of polynomials \(f(x)\in {\mathbb {Z}}[x]\) of any degree \(n\ge 2\), which extend the results of He et al. (Bull Aust Math Soc 82(2):187–204, 2010).
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References
B. He, A. Togbé, M. Ulas, On the Diophantine equation \(z^2=f(x)^2\pm f(y)^2\), II. Bull. Aust. Math. Soc. 82(2), 187–204 (2010)
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S. Tengely, M. Ulas, On certain Diophantine equations of the form \(z^2=f(x)^2pm g(y)^2\). J. Number Theory 174, 239–257 (2017)
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This research was supported by the National Natural Science Foundation of China (Grant No. 11501052), Younger Teacher Development Program of Changsha University of Science and Technology (Grant No. 2019QJCZ051), and Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology)
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Zhang, Y., Tang, Q. On the integer solutions of the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\). Period Math Hung 85, 369–379 (2022). https://doi.org/10.1007/s10998-021-00442-6
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DOI: https://doi.org/10.1007/s10998-021-00442-6