Wilson’s theorem result

The following is a variant of Wilson’s theorem:

Let $p$ be a positive integer and $1\le k\le p-1$. Then:
$p$ is prime if and only if

$$(p-k)!(k-1)!\equiv {(-1)}^k(modp).$$

For particular $k=1,2,\mathrm{\dots }$ one gets nice results.

Title	Wilson’s theorem result
Canonical name	WilsonsTheoremResult
Date of creation	2013-03-22 14:20:44
Last modified on	2013-03-22 14:20:44
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	6
Author	mathwizard (128)
Entry type	Result
Classification	msc 11-00