Prove that $\displaystyle \forall p$ prime and $\displaystyle r\in\mathbb{Z}$ $\displaystyle \exists n\in\mathbb{N}$ for which $\displaystyle \binom{2n}{n}\equiv r\pmod{p}$.

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- Feb 15th 2008, 10:26 AMjames_bondSeems quite hard
Prove that $\displaystyle \forall p$ prime and $\displaystyle r\in\mathbb{Z}$ $\displaystyle \exists n\in\mathbb{N}$ for which $\displaystyle \binom{2n}{n}\equiv r\pmod{p}$.