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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Binomial coefficients and the ring of $p$-adic integers
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by Zhi-Wei Sun and Wei Zhang PDF
Proc. Amer. Math. Soc. 139 (2011), 1569-1577 Request permission

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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2763746