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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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
Posted: February 3, 2006
MathSciNet review: 2213693
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Abstract | References | Similar Articles | Additional Information

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$.

References

  • [C] H. Cohn, Advanced Number Theory, Dover Publ. Inc., New York, 1962. MR 0594936 (82b:12001)
  • [CP] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, New York, 2001. MR 1821158 (2002a:11007)
  • [DSS] K. Dilcher, L. Skula and I. Sh. Slavutskii, Bernoulli numbers, 1713/1990, Queen's Papers in Pure and Appl. Math. 87(1990). The website of the on-line version is http://www.mathstat. dal.ca/$ \tilde {}$dilcher/bernoulli.html. MR 1119305 (92f:11001)
  • [G] A. Granville, Arithmetic properties of binomial coefficients.I. Binomial coefficients modulo prime powers, in: Organic mathematics (Burnaby, BC, 1995), 253-276, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997. MR 1483922 (99h:11016)
  • [GS] A. Granville and Z. W. Sun, Values of Bernoulli polynomials, Pacific J. Math. 172 (1996), 117-137. MR 1379289 (98b:11018)
  • [IR] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Graduate Texts in Math.; 84), 2nd ed., Springer, New York, 1990. MR 1070716 (92e:11001)
  • [L] M. Lerch, Zur Theorie des Fermatschen Quotienten $ (a^{p-1}-1)/p=q(a)$, Math. Ann. 60 (1905), 471-490.
  • [R] P. Ribenboim, The Book of Prime Number Records, Springer, New York, 1988. MR 0931080 (89e:11052)
  • [S1] Z. W. Sun, Products of binomial coefficients modulo $ p^{2}$, Acta Arith. 97 (2001), 87-98. MR 1819624 (2002m:11013)
  • [S2] Z. W. Sun, On the sum $ \sum _{k\equiv r (\operatorname{mod} m)}\binom nk$ and related congruences, Israel J. Math. 128 (2002), 135-156. MR 1910378 (2003d:11026)
  • [S3] Z. W. Sun, General congruences for Bernoulli polynomials, Discrete Math. 262 (2003), 253-276. MR 1951393 (2003m:11037)
  • [W] H. C. Williams, Some formulae concerning the fundamental unit of a real quadratic field, Discrete Math. 92 (1991), 431-440. MR 1140604 (92j:11126)

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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of China
Email: zwsun@nju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08262-1
PII: S 0002-9939(06)08262-1
Received by editor(s): March 4, 2004
Received by editor(s) in revised form: March 6, 2005
Posted: 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 after 28 years from publication.




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