Binomial coefficients and the ring of $p$-adic integers

Authors:
Zhi-Wei Sun and Wei Zhang

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1569-1577

MSC (2010):
Primary 11B65; Secondary 05A10, 11A07, 11S99

DOI:
https://doi.org/10.1090/S0002-9939-2010-10587-7

Published electronically:
October 28, 2010

MathSciNet review:
2763746

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $k>1$ be an integer and let $p$ be a prime. We show that if $p^{a}\leqslant k<2p^{a}$ or $k=p^{a}q+1$ (with $q<p/2$) for some $a=1,2,3,\ldots$, then the set $\{\binom nk: n=0,1,2,\ldots \}$ is dense in the ring $\mathbb {Z}_{p}$ of $p$-adic integers; i.e., it contains a complete system of residues modulo any power of $p$.

- William D. Banks, Florian Luca, Igor E. Shparlinski, and Henning Stichtenoth,
*On the value set of $n!$ modulo $a$ prime*, Turkish J. Math.**29**(2005), no. 2, 169–174. MR**2142292** - C. Cobeli, M. Vâjâitu, and A. Zaharescu,
*The sequence $n!\pmod p$*, J. Ramanujan Math. Soc.**15**(2000), no. 2, 135–154. MR**1754715** - Kenneth Davis and William Webb,
*A binomial coefficient congruence modulo prime powers*, J. Number Theory**43**(1993), no. 1, 20–23. MR**1200804**, DOI https://doi.org/10.1006/jnth.1993.1002 - Moubariz Z. Garaev and Florian Luca,
*Character sums and products of factorials modulo $p$*, J. Théor. Nombres Bordeaux**17**(2005), no. 1, 151–160 (English, with English and French summaries). MR**2152216** - 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** - Richard K. Guy,
*Unsolved problems in number theory*, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR**1299330** - Hong Hu and Zhi-Wei Sun,
*An extension of Lucas’ theorem*, Proc. Amer. Math. Soc.**129**(2001), no. 12, 3471–3478. MR**1860478**, DOI https://doi.org/10.1090/S0002-9939-01-06234-7 - 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** - M. Ram Murty,
*Introduction to $p$-adic analytic number theory*, AMS/IP Studies in Advanced Mathematics, vol. 27, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR**1913413** - Z. W. Sun,
*On sums of primes and triangular numbers*, Journal of Combinatorics and Number Theory**1**(2009), 65–76.

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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

**Wei Zhang**

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

Email:
zhangwei_07@yahoo.com.cn

Received by editor(s):
December 26, 2009

Received by editor(s) in revised form:
May 15, 2010

Published electronically:
October 28, 2010

Additional Notes:
The first author is the corresponding author. He is supported by the National Natural Science Foundation (grant 10871087) and the Overseas Cooperation Fund (grant 10928101) of China

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2010
American Mathematical Society

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