It is known that for $p$ a prime, the following conditions are equivalent:

- ${2p-1\choose p-1}\equiv 1\mod{p^4}$
- $p^3$ divides the numerator of $\sum_{n=1}^{p-1}\frac{1}{n}$
- $p$ divides the numerator of the Bernoulli number $B_{p-3}$ (i.e., $(p,p-3)$ is an irregular pair)

A prime satisfying these conditions is called a Wolstenholme prime; only two are known. It is conjectured that there are infinitely many, and that their density is 0.

Are there known to be infinitely-many primes which are NOT Wolstenholme primes?

This would follow from the conjecture that their are infinitely many regular primes, but it seems that the property of not being a Wolstenholme prime is much weaker than being regular.