Wednesday, 13 March 2019

number theory - Lucas' Theorem and Pascal's Triangle

I have a general question about Lucas' Theorem. Lucas' Theorem says the following:





Theorem (Lucas' Theorem) Let p be a prime number. Write n and k in base p: n=a0+a1+a2p2++adpd and k=b0+b1p+b2p2++bdpd where 0ai,bip1. Then \ \binom{n}{k} \equiv \prod_{i=0}^{d} \binom{a_i}{b_i} \pmod{p}




Another way of viewing this is that in \mathbb{Z}_p, \binom{n}{k} is a product of binomial coefficients.




Question. How does this theorem correspond to the geometric interpretation of Pascal's Triangle? Namely, for any entry in Pascal's triangle which is odd mark a \text{x}. Else leave it blank. So we basically get a fractal like structure. Is Lucas' Theorem just a statement of a way to get extra rows of Pascal's Triangle given that we know 2^n rows?


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