Wolstenholme's Theorem - Wolstenholme Primes

Wolstenholme Primes

A prime p is called a Wolstenholme prime iff the following condition holds:

If p is a Wolstenholme prime, then Glaisher's theorem holds modulo p4. The only known Wolstenholme primes so far are 16843 and 2124679 (sequence A088164 in OEIS); any other Wolstenholme prime must be greater than 109. This result is consistent with the heuristic argument that the residue modulo p4 is a pseudo-random multiple of p3. This heuristic predicts that the number of Wolstenholme primes between K and N is roughly ln ln N − ln ln K. The Wolstenholme condition has been checked up to 109, and the heuristic says that there should be roughly one Wolstenholme prime between 109 and 1024. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, meaning that the congruence holds modulo p5.