Binomial coefficients and quadratic fields

Sun, Zhi-Wei

doi:10.1090/S0002-9939-06-08262-1

Binomial coefficients and quadratic fields
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Proc. Amer. Math. Soc. 134 (2006), 2213-2222 Request permission

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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