Wolstenholme's Theorem

In mathematics, Wolstenholme's theorem states that for a prime number p > 3, the congruence

holds, 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

The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo p2, which holds for all primes p (for p=2 only in the second formulation). The second formulation of Wolstenholme's theorem is due to J. W. L. Glaisher and is inspired by Lucas' theorem.

No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo p4 is called a Wolstenholme prime (see below).

As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers:

(Congruences with fractions make sense, provided that the denominators are coprime to the modulus.) For example, with p=7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7.