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

  1. Apr 24, 2017 #1
    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. jcsd
  3. Apr 24, 2017 #2

    fresh_42

    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
     
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