Wilson quotient

The Wilson quotient W(p) is defined as:

${\displaystyle W(p)={\frac {(p-1)!+1}{p}}}$

If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence A007619 in the OEIS):

W(2) = 1

W(3) = 1

W(5) = 5

W(7) = 103

W(11) = 329891

W(13) = 36846277

W(17) = 1230752346353

W(19) = 336967037143579

...

It is known that[1]

${\displaystyle W(p)\equiv B_{2(p-1)}-B_{p-1}{\pmod {p}},}$

${\displaystyle p-1+ptW(p)\equiv pB_{t(p-1)}{\pmod {p^{2}}},}$

where ${\displaystyle B_{k}}$ is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting ${\displaystyle t=1}$ and ${\displaystyle t=2}$.