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On the Diophantine equation $ x\sp {2n}-\mathcal{D}y^2=1$


Author: Zhen Fu Cao
Journal: Proc. Amer. Math. Soc. 98 (1986), 11-16
MSC: Primary 11D41
DOI: https://doi.org/10.1090/S0002-9939-1986-0848864-4
MathSciNet review: 848864
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Abstract: In this paper, it has been proved that if $ n > 2$ and Pell's equation $ {u^2} - \mathcal{D}{v^2} = - 1$ has integer solution, then the equation $ {x^{2n}} - \mathcal{D}{y^2} = 1$ has only solution in positive integers $ x = 3$, $ y = 22$ (when $ n = 5$, $ \mathcal{D} = 122$). That is proved by studying the equations $ {x^p} + 1 = 2{y^2}$ and $ {x^p} - 1 = 2{y^2}$ ($ p$ is an odd prime). In addition, some applications of the above result are given.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0848864-4
Article copyright: © Copyright 1986 American Mathematical Society

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