The conjecture: $\binom{2n-1}{n-1} \equiv 1 \pmod{n^3} \quad \Longrightarrow \quad n \in \mathbb{P}$

Question

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.

Answer 1

This is known as Wolstenholme's theorem.

For a prime $p > 3$ we have:

$$\binom{2p-1}{p-1} \equiv 1 \pmod{p^3}$$