Wolstenholme's Theorem

Wolstenholme's theorem, proved by Joseph Wolstenholme in 1862, states that for every prime p of at least 5, the binomial coefficient of 2p minus 1 choose p minus 1 is congruent to 1 modulo p cubed, with an equivalent formulation in terms of congruences among generalized harmonic numbers. Charles Babbage had already proved a weaker version in 1819, showing the same congruence modulo p squared for every prime p of at least 3. The theorem gives rise to the Wolstenholme primes, those satisfying the congruence modulo p to the fourth power, of which only two are currently known, 16843 and 2124679, with none found beyond 10 to the 11th power; whether any composite number satisfies the theorem's congruence at all remains an open question.