- Wolstenholme's theorem
In
mathematics , Wolstenholme's theorem states that for aprime number "p" > 3, the congruence:2p-1 choose p-1} equiv 1 , mod , p^3holds, 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:ap choose bp} equiv {a choose b} mod , p^3.The theorem was first proved byJoseph Wolstenholme in 1862;Charles Babbage had shown the equivalent for "p"2 in 1819.No known
composite number s 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 prime s". 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 number congruences: :p-1)!left(1+{1 over 2}+{1 over 3}+...+{1 over p-1} ight) equiv 0 , mod , p^2 mbox{, and}:p-1)!^2left(1+{1 over 2^2}+{1 over 3^2}+...+{1 over (p-1)^2} ight) equiv 0 , mod , 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.ee also
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Fermat's little theorem References
J. Wolstenholme, "On certain properties of prime numbers", Quarterly Journal of Mathematics 5 (1862), pp. 35–39.
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