The Wilson quotient W(p) is defined as:
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
where is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting and .
See also Fermat quotient
References
External links MathWorld: Wilson Quotient