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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
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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 $p$.


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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.