Wolstenholme's Conjecture

Conjecture
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Suppose that:
 * $\dbinom {2 n - 1} {n - 1} \equiv 1 \pmod {n^3}$

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

Then $n$ is a prime number

Also see

 * Wolstenholme's Theorem, of which this is the converse