Wilson quotient

The Wilson quotient W(p) is defined as:

: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 :

: 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

:W(p)\equiv B_{2(p-1)}-B_{p-1}\pmod{p}, :p-1+ptW(p)\equiv pB_{t(p-1)}\pmod{p^2}, where B_k is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting t=1 and t=2.