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

 

Binomial coefficients and quadratic fields


Author: Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 134 (2006), 2213-2222
MSC (2000): Primary 11B65; Secondary 11B37, 11B68, 11R11
Published electronically: February 3, 2006
MathSciNet review: 2213693
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Abstract: Let $ E$ be a real quadratic field with discriminant $ d\not \equiv 0 (\operatorname{mod} p)$ where $ p$ is an odd prime. For $ \rho =\pm 1$ we determine $ \prod_{0<c<d, (\frac dc)=\rho }\binom{p-1} {\lfloor pc/d\rfloor }$ modulo $ p^{2}$ in terms of a Lucas sequence, the fundamental unit and the class number of $ E$.


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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08262-1
PII: S 0002-9939(06)08262-1
Received by editor(s): March 4, 2004
Received by editor(s) in revised form: March 6, 2005
Published electronically: February 3, 2006
Additional Notes: The author was supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.