Wolstenholme's theorem

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Statement

Statement in terms of near-central binomial coefficients

Suppose is a prime number greater than 3. Then, we have the following congruence condition on the binomial coefficient:

Statement in terms of prime cancellation from binomial coefficient

Suppose is a prime number greater than 3. Then, we have the following congruence condition:

This version is due to Glaisher and is similar in structure to Lucas' theorem.

Statement in terms of generalized harmonic numbers

The following two congruences hold:

and

What we mean by this is that if we simplify the expressions on the left side and bring them into fractions in reduced form, the numerators are divisible by and respectively.