Combinatorial congruences modulo prime powers
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- by Zhi-Wei Sun and Donald M. Davis PDF
- Trans. Amer. Math. Soc. 359 (2007), 5525-5553 Request permission
Abstract:
Let $p$ be any prime, and let $\alpha$ and $n$ be nonnegative integers. Let $r\in \mathbb {Z}$ and $f(x)\in \mathbb {Z}[x]$. We establish the congruence \begin{equation*}p^{\deg f}\sum _{k\equiv r (\operatorname {mod}p^{\alpha })} \binom nk(-1)^{k}f\left (\frac {k-r}{p^{\alpha }}\right ) \equiv 0\ \left (\operatorname {mod} p^{\sum _{i=\alpha }^{\infty } \lfloor n/{p^{i}}\rfloor }\right )\end{equation*} (motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas’ theorem: If $\alpha$ is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then \begin{equation*}\begin {split} &\frac {1}{\lfloor n/p^{\alpha -1}\rfloor !} \sum _{k\equiv r (\operatorname {mod}p^{\alpha })}\binom {pn+s}{pk+t}(-1)^{pk} \left (\frac {k-r}{p^{\alpha -1}}\right )^{l}\\ \equiv & \frac {1}{\lfloor n/p^{\alpha -1}\rfloor !} \sum _{k\equiv r (\operatorname {mod} p^{\alpha })}\binom nk\binom st(-1)^{k} \left (\frac {k-r}{p^{\alpha -1}}\right )^{l} \ (\operatorname {mod} p). \end{split}\end{equation*} We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a $p$-adic order bound given by the authors in a previous paper can be attained when $p=2$.References
- D. F. Bailey, Two $p^3$ variations of Lucas’ theorem, J. Number Theory 35 (1990), no. 2, 208–215. MR 1057323, DOI 10.1016/0022-314X(90)90113-6
- P. Colmez, Une correspondance de Langlands locale $p$-adique pour les représentations semi-stables de dimension 2, preprint, 2004.
- D. M. Davis and Z. W. Sun, A number-theoretic approach to homotopy exponents of SU$(n)$, J. Pure Appl. Algebra 209 (2007), 57–69.
- L. E. Dickson, History of the Theory of Numbers, Vol. I, AMS Chelsea Publ., 1999.
- 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
- 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
- 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, DOI 10.1007/978-1-4757-2103-4
- J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd ed., Cambridge University Press, Cambridge, 2001. MR 1871828, DOI 10.1017/CBO9780511987045
- Zhi-Wei Sun, On the sum $\sum _{k\equiv r\pmod m}{n\choose k}$ and related congruences, Israel J. Math. 128 (2002), 135–156. MR 1910378, DOI 10.1007/BF02785421
- Zhi-Wei Sun, General congruences for Bernoulli polynomials, Discrete Math. 262 (2003), no. 1-3, 253–276. MR 1951393, DOI 10.1016/S0012-365X(02)00504-6
- Zhi-Wei Sun, Polynomial extension of Fleck’s congruence, Acta Arith. 122 (2006), no. 1, 91–100. MR 2217327, DOI 10.4064/aa122-1-9
- Daqing Wan, Combinatorial congruences and $\psi$-operators, Finite Fields Appl. 12 (2006), no. 4, 693–703. MR 2257090, DOI 10.1016/j.ffa.2005.08.006
- Carl S. Weisman, Some congruences for binomial coefficients, Michigan Math. J. 24 (1977), no. 2, 141–151. MR 463093
Additional Information
- Zhi-Wei Sun
- Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
- MR Author ID: 254588
- Email: zwsun@nju.edu.cn
- Donald M. Davis
- Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
- MR Author ID: 55085
- Email: dmd1@lehigh.edu
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 5525-5553
- MSC (2000): Primary 11B65; Secondary 05A10, 11A07, 11B68, 11S05
- DOI: https://doi.org/10.1090/S0002-9947-07-04236-5
- MathSciNet review: 2327041