Let p be a composite, and let us apply the simple rule stated by Lucas' theorem to check whether p is prime (that is, involving the p base representation). Now, assume that the result is positive; would that mean that indeed p divides $n \choose k$, or the result is random, since the hypothesis of p being a prime is violated?
On Lucas' Theorem over composite
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combinatorics
primality-test
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2What do you mean by "the simple rule stated by Lucas' theorem"? (I know what you mean, but others might not.) – 2012-06-06
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5"Let $p$ be composite" --- words I thought I'd never see. – 2012-06-06
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0You are asking whether the congruence stated in Lucas's Theorem holds if and only if the modulus is prime, that is, whether Lucas's Theorem can be used as a primality test, the way Wilson's Theorem can? – 2012-06-07
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0@ArturoMagidin He seems to only refer to the special case of Lucas' Theorem where the binomial coefficient is zero, but I guess Qiaochu can help us here after claiming "I know what you mean". – 2012-06-09