Are there infinitely many Wolstenholme primes

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.

A Wolstenholme prime is a prime $p\ge5$ such that $\binom{2p-1}{p-1}\equiv1\pmod{p^4}$, where $\binom ab=a!/(b!(a-b)!)$. This is stronger than Wolstenholme's congruence modulo $p^3$ valid for every prime $p\ge5$.

Statement recorded in the supplied source.

Are there infinitely many Wolstenholme primes? [S2]

Short English statement

Do infinitely many primes satisfy the binomial congruence with a fourth power modulus?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400166 (AMR-093-0166). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Are there infinitely many Wolstenholme primes. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 Jonathan Sondow, Extending Babbage's (non-)primality tests, arXiv:1812.07650 (2018) Equation (10) and Wolstenholme-prime discussion, pp. 4--5. https://arxiv.org/pdf/1812.07650.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A088164: Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4)., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A088/A088164.seq.