# Wilson’s theorem result

The following is a variant of Wilson’s theorem:

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

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

For particular $k=1,2,\ldots$ one gets nice results.

Title Wilson’s theorem result WilsonsTheoremResult 2013-03-22 14:20:44 2013-03-22 14:20:44 mathwizard (128) mathwizard (128) 6 mathwizard (128) Result msc 11-00