 Wolstenholme's theorem

For every prime $p\ge 5$, $\left(\genfrac{}{}{0ex}{}{2p-1}{p-1}\right)\equiv 1(\mathrm{mod}{p}^3)$. Equivalently, ${\sum }_{k=1}^{p-1}1/k\equiv 0(\mathrm{mod}{p}^2)$, with denominators interpreted through modular inverses.