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

 

Integer parts of powers of quadratic units


Author: Daniel Cass
Journal: Proc. Amer. Math. Soc. 101 (1987), 610-612
MSC: Primary 11R11; Secondary 11B05, 11R27
MathSciNet review: 911018
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Abstract: Let $ \alpha > 1$ be a unit in a quadratic field. The integer part of $ {\alpha ^n}$, denoted $ [{\alpha ^n}]$, is shown to be composite infinitely often. Provided $ \alpha \ne (1 + \sqrt 5 )/2$, it is shown that the number of primes among $ [\alpha ],[{\alpha ^2}], \ldots ,[{\alpha ^n}]$ is bounded by a function asymptotic to $ c \cdot {\log ^2}n$, with $ c = 1/(2\log 2 \cdot \log 3)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0911018-8
PII: S 0002-9939(1987)0911018-8
Keywords: Quadratic unit, integer part
Article copyright: © Copyright 1987 American Mathematical Society