Binomial coefficients modulo a prime $p>5$ Can we prove that for every prime $p > 5$ the equality
$\binom {2p}{p} \equiv 2\mod p^3$ holds.

Did you check it for even one prime $p > 5$? It is already wrong for $p = 7$...
What we can prove, however, is
$${2p \choose p} \equiv 2 \mod{p^3}.$$
You can find more about it on this Wikipedia page.
