The Wilson quotient $W_n$ for a given positive integer $n$ is the rational number $\displaystyle \frac{\Gamma(n) + 1}{n}$ , where $\Gamma(x)$ is Euler's Gamma function (since we're dealing with integer inputs here, in effect this is merely a quicker way to write $(n - 1)!$ ).

From Wilson's theorem it follows that the Wilson quotient is an integer only if $n$ is not composite. When $n$ is composite, the numerator of the Wilson quotient is $(n - 1)! + 1$ and the denominator is $n$ . For example, if $n = 7$ we have numerator 721 with denominator 7, and since these have 7 as their greatest common divisor, in lowest terms the Wilson quotient of 7 is 103 (with 1 as tacit numerator). But for $n = 8$ we have $$W_8 = \frac{5041}{8}.$$

Bibliography

1

R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective. New York: Springer (2001): 29.