Open Access
Sept. 2000 Some congruences for binomial coefficients. II
Sang Geun Hahn, Dong Hoon Lee
Proc. Japan Acad. Ser. A Math. Sci. 76(7): 104-107 (Sept. 2000). DOI: 10.3792/pjaa.76.104


Let $t \equiv 3 \mod 4$ ($t > 3$) be a prime and $\sigma_r\colon \zeta_t \rightarrow \zeta_t^r$ be a generator of $\operatorname{Gal}(\mathbf{Q}(\zeta_t) / \mathbf{Q}(\sqrt{-t}))$ for $r \in \{1,\dots,t-1\}$. If $p = tn + r$ is a prime, then $4p^h$ can be expressed as the form $4p^h = a^2 + tb^2$ where $h$ is the class number of $\mathbf{Q}(\sqrt{-t})$. Let $\alpha t$ be the sum of representatives of $\langle r \rangle $ in $(\mathbf{Z}/t\mathbf{Z})^{\times}$ and $\beta = \phi(t)/2 - \alpha$. If we choose the sign of $a$ then $a \equiv 2p^{\beta} \mod t$ and $a$ satisfies a certain congruence relation modulo $p$. We also treat the case of $t = 4k$ for a prime $k \equiv 1 \mod 4$.


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Sang Geun Hahn. Dong Hoon Lee. "Some congruences for binomial coefficients. II." Proc. Japan Acad. Ser. A Math. Sci. 76 (7) 104 - 107, Sept. 2000.


Published: Sept. 2000
First available in Project Euclid: 23 May 2006

zbMATH: 0971.11046
MathSciNet: MR1785634
Digital Object Identifier: 10.3792/pjaa.76.104

Primary: 11L05
Secondary: 11A07 , 11B65 , 11T24

Keywords: binomial coefficients , Eisenstein sum , Gauss sum

Rights: Copyright © 2000 The Japan Academy

Vol.76 • No. 7 • Sept. 2000
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