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# Wilson quotient

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

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

# References

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

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## Mathematics Subject Classification

11A51*no label found*11A41

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