Wolstenholme's theorem

In mathematics, Wolstenholme's theorem states that for a prime number ${\displaystyle p\geq 5}$, the congruence

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

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. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo p², which holds for ${\displaystyle p\geq 3}$. An equivalent formulation is the congruence

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

for ${\displaystyle p\geq 5}$, which is due to Wilhelm Ljunggren^[1] (and, in the special case ${\displaystyle b=1}$, to J. W. L. Glaisher^{[citation needed]}) 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:

${\displaystyle 1+{1 \over 2}+{1 \over 3}+\dots +{1 \over p-1}\equiv 0{\pmod {p^{2}}}{\mbox{, and}}}$

${\displaystyle 1+{1 \over 2^{2}}+{1 \over 3^{2}}+\dots +{1 \over (p-1)^{2}}\equiv 0{\pmod {p}}.}$

since

${\displaystyle {2p-1 \choose p-1}=\prod _{1\leq k\leq p-1}{\frac {2p-k}{k}}\equiv 1-2p\sum _{1\leq k\leq p-1}{\frac {1}{k}}{\pmod {p^{2}}}}$

(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.