Wolstenholme prime can be defined in a number of equivalent ways.
Definition via binomial coefficients
A Wolstenholme prime is a prime number p > 7 that satisfies the congruence
where the expression in left-hand side denotes a binomial coefficient.
Compare this with Wolstenholme's theorem, which states that for every prime p > 3 the following congruence holds:
Definition via Bernoulli numbers
A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3. The Wolstenholme primes therefore form a subset of the irregular primes.
Definition via irregular pairs
A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.
Definition via harmonic numbers
A Wolstenholme prime is a prime p such that
i.e. the numerator of the harmonic number expressed in lowest terms is divisible by p3.