- Wolstenholme's theorem
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 Wolstenholmein 1862; Charles Babbagehad shown the equivalent for "p"2 in 1819.
composite numbers satisfy Wolstenholme's theorem. Very few prime numbers satisfy the equivalent for "p"4: 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 "p"4 is a pseudo-random multiple of "p"3. 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 numbercongruences: ::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.
Fermat's little theorem
J. Wolstenholme, "On certain properties of prime numbers", Quarterly Journal of Mathematics 5 (1862), pp. 35–39.
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