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Binomial coefficients and the ring of p-adic integers

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arxiv 0812.3089 v5 pith:UBDXTOL5 submitted 2008-12-16 math.NT math.CO

Binomial coefficients and the ring of p-adic integers

classification math.NT math.CO
keywords integersp-adicringbinombinomialcoefficientscompletecontains
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.

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