The conjecture: $\binom{2n-1}{n-1} \equiv 1 \pmod{n^3} \quad \Longrightarrow \quad n \in \mathbb{P}$ I recall seeing the following conjecture somewhere, but I cannot find the reference any more. Where can I find more information about this conjecture? Does it have a name?
Conjecture: For any natural number $n$ it holds that
$$
\binom{2n-1}{n-1} \equiv 1 \pmod{n^3} \quad \Longrightarrow \quad n \in \mathbb{P},
$$
where $\mathbb{P}$ denotes the set of primes.

This is known as Wolstenholme's theorem.
For a prime $p > 3$ we have:
$$\binom{2p-1}{p-1} \equiv 1 \pmod{p^3}$$
