Wolstenholme's Theorem

Theorem
Let $p$ be a prime number such that $p \ge 5$.

Then:
 * $\dbinom {2 p - 1} {p - 1} \equiv 1 \pmod {p^3}$

where $\dbinom {2 p - 1} {p - 1}$ denotes a binomial coefficient.

Also see

 * Wolstenholme's Conjecture


 * Babbage's Congruence