Wolstenholme's theorem
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In mathematics, Wolstenholme's theorem states that for a prime number p > 3, the congruence
holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343. An equivalent formulation is the congruence
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:
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.
[edit] References
J. Wolstenholme, "On certain properties of prime numbers", Quarterly Journal of Mathematics 5 (1862), pp. 35–39.