High Quality Content by WIKIPEDIA articles! The theorem was first proved by Joseph Wolstenholme in 1862; Charles Babbage had shown the equivalent for p2 in 1819. The second formulation is related to Lucas' theorem.No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none. A prime that satisfies the congruence modulo p4 is called a "Wolstenholme prime".The only known Wolstenholme primes so far are 16843 and 2124679,any other Wolstenholme prime must be greater than 109.This data 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.

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