Wolstenholme's Theorem

If  is a prime , then the numerator of the harmonic number

(1)

is divisible by  and the numerator of the generalized harmonic number

(2)

is divisible by . The numerators of  are sometimes known as Wolstenholme numbers.

These imply that if  is prime, then

(3)

REFERENCES:

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 85, 1994.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 21, 1989.