Binomial coefficients and quadratic fields
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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 $E$.References
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Additional Information
- Zhi-Wei Sun
- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- MR Author ID: 254588
- Email: zwsun@nju.edu.cn
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2213-2222
- MSC (2000): Primary 11B65; Secondary 11B37, 11B68, 11R11
- DOI: https://doi.org/10.1090/S0002-9939-06-08262-1
- MathSciNet review: 2213693