# 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 Bp-3, where Bn is the nth Bernoulli number.