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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on the Diophantine equation $ x\sp {2p}-Dy\sp 2=1$


Author: Mao Hua Le
Journal: Proc. Amer. Math. Soc. 107 (1989), 27-34
MSC: Primary 11D25; Secondary 11D41
MathSciNet review: 965245
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Abstract: Let $ D$ be a positive integer which is square free, and let $ p$ be a prime. In this note we prove that if $ p = 2$ and $ D > \exp 64$ , then the equation $ {x^{2p}} - D{y^2} = 1$ has at most one positive integer solution $ \left( {x,y} \right)$; if $ p > 2$ and $ D > \exp \exp \exp \exp 10$, then every positive integer solution $ \left( {x,y} \right)$ can be expressed as $ {x^p} + y\sqrt D = \varepsilon _1^m$, where $ m$ is a positive integer with $ 2\nmid m,{\varepsilon _1}$ is the fundamental solution of Pell's equation $ {u^2} - D{v^2} = 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0965245-6
PII: S 0002-9939(1989)0965245-6
Article copyright: © Copyright 1989 American Mathematical Society