Wolstenholme prime

Wilson's theorem can be used to show that the binomial coefficient (np-1 choose p-1) is one modulo p for all primes p and all integers n.  In 1819 Babbage noticed that (2p-1 choose p-1) is one modulo p² for all odd primes.  In 1862, Wolstenholme improved this by proving that (2p-1 choose p-1) is one modulo p³ 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 p⁴, 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/(p-1)

is divisible by p² and the numerator of

1 + 1/2² + 1/3² + ... + 1/(p-1)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:

• The primes p for which the central binomial coefficient (2p choose p) is 2 modulo p⁴.
• The primes p > 7 for which the sum from k = floor(p/6) + 1 to floor(p/4) of 1/k³ is divisible by p.
• The primes p which divide B_p-3, where B_n is the nth Bernoulli number.

See Also: WilsonPrime, WieferichPrime

Related pages (outside of this work)

• Arithmetic properties of Binomial Coefficients

References:

Apostol76 (p. 116)

T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York, NY, 1976.  pp. xii+338, ISBN 0-387-90163-9. MR 55:7892 [QA241.A6]

HW79 (p. 88-89)

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford University Press, 1979.  ISBN 0198531702. MR 81i:10002 (Annotation available)

Ribenboim95 (p. 29)

P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]