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Wolstenholme's TheoremIn mathematics, Wolstenholme's theorem states that for a prime number p > 3, the congruence -
holds, where the LHS is a binomial coefficient. 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. Wolstenholme's theorem can be broken down into two other results: -
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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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