Wolstenholme's theorem

Statement

Statement in terms of near-central binomial coefficients

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

${\binom {2p-1}{p-1}}\equiv 1{\pmod {p^{3}}}$

Statement in terms of prime cancellation from binomial coefficient

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

${\binom {ap}{bp}}\equiv {\binom {a}{b}}{\pmod {p^{3}}}$

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:

$\!1+{\frac {1}{2}}+{\frac {1}{3}}+\dots +{\frac {1}{p-1}}\equiv 0{\pmod {p^{2}}}$

and

$\!1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{(p-1)^{2}}}\equiv 0{\pmod {p}}$

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 $p^{2}$ and $p$ respectively.