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Math Help - Proof

  1. #1
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    Proof

    Prove that:

    2p choose p is congruent to 2 modulus p^2

    where p is any prime number
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  2. #2
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    Quote Originally Posted by Kaiserdivus View Post
    Prove that:

    2p choose p is congruent to 2 modulus p^2

    where p is any prime number
    in general we have \binom{2n}{n} = \sum_{k=0}^n \binom{n}{k}^2 = 2 + \sum_{k=1}^{n-1} \binom{n}{k}^2. now if n = p is prime, then since p \mid \binom{p}{k}, \ 1 \leq k \leq p-1, we'll have: \binom{2p}{p} = 2 + \sum_{k=1}^{p-1} \binom{p}{k}^2 \equiv 2 \mod p^2. \ \ \ \square
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