Wolstenholme's converse problem

Prove that every integer n>3 satisfying n^3 divides the binomial coefficient expression \(\binom{2n-1}{n-1}-1\) is prime, thereby establishing the converse of Wolstenholme's theorem and characterizing the prime numbers.

Background

The paper presents Wolstenholme's theorem, which states that for every prime p>3, p3∣(2pp)−2p^3\mid\binom{2p}{p}-2. It then formulates the converse problem: determine whether the divisibility condition n3∣(2n−1n−1)−1n^3\mid\binom{2n-1}{n-1}-1 forces an integer n>3 to be prime.

This conjecture is identified as Wolstenholme's converse problem and is attributed to J. P. Jones. The paper notes that resolving it affirmatively would characterize the prime numbers, while also reporting that computational verification had established the claim for all n<109n<10^9.

References

If $n>3$ and $n3\mid\binom{2n-1}{n-1}-1$ then $n$ is a prime number. The Conjecture \ref{conj01} is known as Wolstenholme's converse problem which if true will characterize the prime numbers. This conjecture was proposed by J. P. Jones and remains open.

— Perfect numbers, Wieferich primes and the solutions of $\binom{2n}{n}\equiv 2^n \bmod n$  (2609.03091 - Guedes et al., 2 Sep 2026) in Section 1, Introduction, Conjecture 1