The Wilson quotient for a given positive integer is the rational number , where is Euler's Gamma function (since we're dealing with integer inputs here, in effect this is merely a quicker way to write ).

From Wilson's theorem it follows that the Wilson quotient is an integer only if is not composite. When is composite, the numerator of the Wilson quotient is and the denominator is . For example, if 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 we have

Bibliography

1

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