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Wilson's theorem can be used to show that the binomial
coefficient (np1 choose p1) is one
modulo p for all primes p and all integers n.
In 1819 Babbage noticed that (2p1 choose p1) is one modulo p^{2} for all odd primes. In 1862, Wolstenholme improved this by
proving that (2p1 choose p1) is
one modulo p^{3} for primes p > 3. It is still unknown if the converse is true.
For a few select primes this congruence is is also true modulo p^{4}, such primes are called the Wolstenholme primes. After searching through all primes up to 500,000,000, the only known Wolstenholme primes remain the lonely pair 16843 and 2124679. Wolstenholme's theorem can be stated in many ways, including that for every prime p > 5 the numerator of 1 + 1/2 + 1/3 + ... + 1/(p1)is divisible by p^{2} and the numerator of 1 + 1/2^{2} + 1/3^{2} + ... + 1/(p1)^{2}is divisible by p. There are many similar results, see the binomial coefficient web site linked below for most of them. Other was to characterize the Wolstenholme primes include:
See Also: WilsonPrime, WieferichPrime Related pages (outside of this work) References:
Chris K. Caldwell © 19992014 (all rights reserved)
