Wilson quotient

The Wilson quotientMathworldPlanetmath Wn for a given positive integer n is the rational number Γ(n)+1n, where Γ(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 divisorMathworldPlanetmathPlanetmath, in lowest terms the Wilson quotient of 7 is 103 (with 1 as tacit numerator). But for n=8 we have



  • 1 R. Crandall & C. Pomerance, Prime NumbersMathworldPlanetmath: 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