Binomial coefficients and quadratic fields

Author:
Zhi-Wei Sun

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2213-2222

MSC (2000):
Primary 11B65; Secondary 11B37, 11B68, 11R11

Published electronically:
February 3, 2006

MathSciNet review:
2213693

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real quadratic field with discriminant where is an odd prime. For we determine modulo in terms of a Lucas sequence, the fundamental unit and the class number of .

**[C]**Harvey Cohn,*Advanced number theory*, Dover Publications, Inc., New York, 1980. Reprint of A second course in number theory, 1962; Dover Books on Advanced Mathematics. MR**594936****[CP]**Richard Crandall and Carl Pomerance,*Prime numbers*, Springer-Verlag, New York, 2001. A computational perspective. MR**1821158****[DSS]**Karl Dilcher, Ladislav Skula, and Ilja Sh. Slavutskiǐ,*Bernoulli numbers*, Queen’s Papers in Pure and Applied Mathematics, vol. 87, Queen’s University, Kingston, ON, 1991. Bibliography (1713–1990). MR**1119305****[G]**Andrew Granville,*Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers*, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. MR**1483922****[GS]**Andrew Granville and Zhi-Wei Sun,*Values of Bernoulli polynomials*, Pacific J. Math.**172**(1996), no. 1, 117–137. MR**1379289****[IR]**Kenneth Ireland and Michael Rosen,*A classical introduction to modern number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR**1070716****[L]**M. Lerch,*Zur Theorie des Fermatschen Quotienten*, Math. Ann.**60**(1905), 471-490.**[R]**Paulo Ribenboim,*The book of prime number records*, Springer-Verlag, New York, 1988. MR**931080****[S1]**Zhi-Wei Sun,*Products of binomial coefficients modulo 𝑝²*, Acta Arith.**97**(2001), no. 1, 87–98. MR**1819624**, 10.4064/aa97-1-5**[S2]**Zhi-Wei Sun,*On the sum ∑_{𝑘≡𝑟\pmod𝑚}𝑛\choose𝑘 and related congruences*, Israel J. Math.**128**(2002), 135–156. MR**1910378**, 10.1007/BF02785421**[S3]**Zhi-Wei Sun,*General congruences for Bernoulli polynomials*, Discrete Math.**262**(2003), no. 1-3, 253–276. MR**1951393**, 10.1016/S0012-365X(02)00504-6**[W]**H. C. Williams,*Some formulas concerning the fundamental unit of a real quadratic field*, Discrete Math.**92**(1991), no. 1-3, 431–440. MR**1140604**, 10.1016/0012-365X(91)90298-G

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11B65,
11B37,
11B68,
11R11

Retrieve articles in all journals with MSC (2000): 11B65, 11B37, 11B68, 11R11

Additional Information

**Zhi-Wei Sun**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
zwsun@nju.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-06-08262-1

Received by editor(s):
March 4, 2004

Received by editor(s) in revised form:
March 6, 2005

Published electronically:
February 3, 2006

Additional Notes:
The author was supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.