Wilson quotient
The Wilson quotient^{} ${W}_{n}$ for a given positive integer $n$ is the rational number $\frac{\mathrm{\Gamma}(n)+1}{n}$, where $\mathrm{\Gamma}(x)$ 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 |
---|---|
Canonical name | WilsonQuotient |
Date of creation | 2013-03-22 17:57:47 |
Last modified on | 2013-03-22 17:57:47 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 6 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 11A51 |
Classification | msc 11A41 |