I What is a good way to introduce Wilson's Theorem?

1. Apr 24, 2017

matqkks

What is the most motivating way to introduce Wilson’s Theorem? Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier but apart from working with a particular (p-1) factorial of a prime what other reasons are there for Wilson’s theorem to be useful?

Are there any good resources on this topic?

2. Apr 24, 2017

Staff: Mentor

I doubt that there is a practical use. However, it is a quite funny result, especially if formulated as
$$(n-1)! \equiv \begin{cases} -1 \;(\operatorname{mod} n)& \textrm{ if } n \textrm{ prime }\\ 2 \;(\operatorname{mod} n)& \textrm{ if } n =4\\ 0 \;(\operatorname{mod} n)& \textrm{ other cases }\end{cases}$$
or elegant as $(p-1)! \equiv -1\; (\operatorname{mod} p) \Leftrightarrow p \textrm{ prime }$.

I also find the historical part interesting as Wilson only re-discovered it 700 years later:
https://en.wikipedia.org/wiki/Ibn_al-Haytham#Number_theory