TITLE: Is there a combinatorial proof of this congruence identity?
QUESTION [5 upvotes]: Prove that
$$\binom{2p}{p} \equiv 2\pmod{p^3},$$
where $p\ge 5$ is a prime number.

REPLY [8 votes]: A combinatorial proof for the congruence $\bmod p^2$ is given at this MO question but the answerer suggests the congruence $\bmod p^3$ does not have a natural combinatorial proof.