Wolstenholme's theorem

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In mathematics, Wolstenholme's theorem states that for a prime number p > 3, the congruence

{2p-1 \choose p-1} \equiv 1 \, \bmod \, p^3

holds, where the LHS is a binomial coefficient. An equivalent formulation is the congruence

{ap \choose bp} \equiv {a \choose b} \bmod \, p^3

For example, with p = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862; Charles Babbage had shown the equivalent for p2 in 1819.

No known composite numbers satisfy Wolstenholme's theorem. Very few prime numbers satisfy the equivalent for p4: the two known values that do, 16843 and 2124679, are called Wolstenholme primes. The existing Wolstenholme primes are consistent with the heuristic that the residue modulo p4 is a pseudo-random multiple of p3. This heuristic predicts that the number of Wolstenholme primes less than N is roughly ln ln N.

Wolstenholme's theorem can also be expressed as a pair of Bernoulli number congruences:

(p-1)!\left(1+{1 \over 2}+{1 \over 3}+...+{1 \over p-1}\right) \equiv 0 \, \bmod \, p^2 \mbox{, and}
(p-1)!^2\left(1+{1 \over 2^2}+{1 \over 3^2}+...+{1 \over (p-1)^2}\right) \equiv 0 \, \bmod \, p.

For example, with p=7, the first of these says that 1764 is a multiple of 49, while the second says 773136 is a multiple of 7.

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