On the Diophantine equation $x^ {2n}-\mathcal {D}y^2=1$
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- by Zhen Fu Cao PDF
- Proc. Amer. Math. Soc. 98 (1986), 11-16 Request permission
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.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- 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