## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Some congruences for binomial coefficients. II

#### 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

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

#### 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

#### References

• Berndt, B. C., Evans, R. J., and Williams, K. S.: Gauss and Jacobi Sums. Canad. Math. Soc. Ser. Monogr. Adv. Texts, vol. 21, Wiley, New York (1998).
• Clarke, F.: Closed formulas for representing primes by binary quadratic forms (1998) (preprint, announced in ICM, Berlin).
• Cohen, H.: A Course in Computational Algebraic Number Theory. Grad. Texts in Math., vol. 138, Springer, Berlin-Heidelberg-New York (1995).
• Eisenstein, G.: Zur Theorie der quadratischen Zerfällung der Primzahlen $8n+3$, $7n+2$ und $7n+\nobreak4$. J. Reine Angew. Math., 37, 97–126 (1848).
• Hahn, S., and Lee, D. H.: Some congruences for binomial coefficients. Proc. for Class Field Theory - its Centenary and Prospect (1998) (to appear).
• Lang, S.: Cyclotomic Fields I and II. 2nd ed., Grad. Texts in Math., vol. 121, Springer, Berlin-Heidelberg-New York (1990).
• Lee, D. H., and Hahn, S.: Gauss sums and binomial coefficients. J. Number Theory (2000) (submitted).
• Washington, L. C.: Introduction to Cyclotomic Fields. 2nd ed., Grad. Texts in Math., vol. 83, Springer, Berlin-Heidelberg-New York (1997).