Combinatorial congruences modulo prime powers

Authors:
Zhi-Wei Sun and Donald M. Davis

Journal:
Trans. Amer. Math. Soc. **359** (2007), 5525-5553

MSC (2000):
Primary 11B65; Secondary 05A10, 11A07, 11B68, 11S05

Published electronically:
May 1, 2007

MathSciNet review:
2327041

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence

**[B]**D. F. Bailey,*Two 𝑝³ variations of Lucas’ theorem*, J. Number Theory**35**(1990), no. 2, 208–215. MR**1057323**, 10.1016/0022-314X(90)90113-6**[C]**P. Colmez,*Une correspondance de Langlands locale -adique pour les représentations semi-stables de dimension*2, preprint, 2004.**[DS]**D. M. Davis and Z. W. Sun,*A number-theoretic approach to homotopy exponents of*SU, J. Pure Appl. Algebra**209**(2007), 57-69.**[D]**L. E. Dickson,*History of the Theory of Numbers,*Vol. I, AMS Chelsea Publ., 1999.**[GKP]**Ronald L. Graham, Donald E. Knuth, and Oren Patashnik,*Concrete mathematics*, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR**1397498****[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****[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****[LW]**J. H. van Lint and R. M. Wilson,*A course in combinatorics*, 2nd ed., Cambridge University Press, Cambridge, 2001. MR**1871828****[S02]**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**[S03]**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**[S06]**Zhi-Wei Sun,*Polynomial extension of Fleck’s congruence*, Acta Arith.**122**(2006), no. 1, 91–100. MR**2217327**, 10.4064/aa122-1-9**[W]**Daqing Wan,*Combinatorial congruences and 𝜓-operators*, Finite Fields Appl.**12**(2006), no. 4, 693–703. MR**2257090**, 10.1016/j.ffa.2005.08.006**[We]**Carl S. Weisman,*Some congruences for binomial coefficients*, Michigan Math. J.**24**(1977), no. 2, 141–151. MR**0463093**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11B65,
05A10,
11A07,
11B68,
11S05

Retrieve articles in all journals with MSC (2000): 11B65, 05A10, 11A07, 11B68, 11S05

Additional Information

**Zhi-Wei 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/S0002-9947-07-04236-5

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.