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Binomial coefficients and quadratic fields
Authors:Zhi-Wei Sun
Institution:Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
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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