Combinatorial congruences modulo prime powers
Authors:
ZhiWei Sun and Donald M. Davis
Journal:
Trans. Amer. Math. Soc. 359 (2007), 55255553
MSC (2000):
Primary 11B65; Secondary 05A10, 11A07, 11B68, 11S05
Published electronically:
May 1, 2007
MathSciNet review:
2327041
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence (motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a adic order bound given by the authors in a previous paper can be attained when .
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Arith. 122 (2006), no. 1, 91–100. MR 2217327
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 P. Colmez, Une correspondance de Langlands locale adique pour les représentations semistables de dimension 2, preprint, 2004.
 [DS]
 D. M. Davis and Z. W. Sun, A numbertheoretic approach to homotopy exponents of SU, J. Pure Appl. Algebra 209 (2007), 5769.
 [D]
 L. E. Dickson, History of the Theory of Numbers, Vol. I, AMS Chelsea Publ., 1999.
 [GKP]
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 A. Granville, Arithmetic properties of binomial coefficients.I. Binomial coefficients modulo prime powers, in: Organic Mathematics (Burnaby, BC, 1995), 253276, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997. MR 1483922 (99h:11016)
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 [S03]
 Z. W. Sun, General congruences for Bernoulli polynomials, Discrete Math. 262 (2003), 253276. MR 1951393 (2003m:11037)
 [S06]
 Z. W. Sun, Polynomial extension of Fleck's congruence, Acta Arith. 122 (2006), 91100. MR 2217327 (2007d:11019)
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Additional Information
ZhiWei Sun
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email:
zwsun@nju.edu.cn
Donald M. Davis
Affiliation:
Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
Email:
dmd1@lehigh.edu
DOI:
http://dx.doi.org/10.1090/S0002994707042365
PII:
S 00029947(07)042365
Received by editor(s):
September 6, 2005
Received by editor(s) in revised form:
November 26, 2005
Published electronically:
May 1, 2007
Additional Notes:
The first author is responsible for communications, and partially supported by the National Science Fund for Distinguished Young Scholars (Grant No. 10425103) in People’s Republic of China.
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
