# Wilson quotient

The $W_{n}$ 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

 $W_{8}=\frac{5041}{8}.$

## References

• 1 R. Crandall & C. Pomerance, Prime Numbers: A Computational Perspective. New York: Springer (2001): 29.
Title Wilson quotient WilsonQuotient 2013-03-22 17:57:47 2013-03-22 17:57:47 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11A51 msc 11A41