Proceedings of the Japan Academy, Series A, Mathematical Sciences

Some congruences for binomial coefficients. II

Sang Geun Hahn and Dong Hoon Lee

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Abstract

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 $k \equiv 1 \mod 4$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 76, Number 7 (2000), 104-107.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148393468

Digital Object Identifier
doi:10.3792/pjaa.76.104

Mathematical Reviews number (MathSciNet)
MR1785634

Zentralblatt MATH identifier
0971.11046

Subjects
Primary: 11L05: Gauss and Kloosterman sums; generalizations
Secondary: 11T24: Other character sums and Gauss sums 11A07: Congruences; primitive roots; residue systems 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]

Keywords
Gauss sum Eisenstein sum binomial coefficients

Citation

Lee, Dong Hoon; Hahn, Sang Geun. Some congruences for binomial coefficients. II. Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 7, 104--107. doi:10.3792/pjaa.76.104. https://projecteuclid.org/euclid.pja/1148393468


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