Does the converse of Wolstenholme's theorem hold for all natural numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Wolstenholme's theorem states that every prime $p\ge5$ satisfies $\binom{2p-1}{p-1}\equiv1\pmod{p^3}$, where $\binom ab=a!/(b!(a-b)!)$. Its proposed converse has the same domain of integers at least five.

Statement recorded in the supplied source.

For every integer $n\ge5$, does $\binom{2n-1}{n-1}\equiv1\pmod{n^3}$ imply that $n$ is prime? Equivalently, does every composite integer $n\ge5$ fail this congruence? [S2]

Short English statement

Does the cubic-modulus binomial congruence characterize primes in the range of Wolstenholme’s theorem?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400137 (AMR-093-0137). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jonathan Sondow, Extending Babbage's (non-)primality tests, arXiv:1812.07650 (2018) Equation (10) and converse discussion, pp. 4--5. https://arxiv.org/pdf/1812.07650.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Does the converse of Wolstenholme's theorem hold for all natural numbers. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.