I (probably re)invented a very short combinatorial proof of Wilson's theorem that I perennially teach my students. If it occurs in the literature and someone can tell me where first (or even at all), I would like to attribute credit properly.


I actually prove $p! - p(p-1) \equiv 0 \mod p^2$. 


$p!$ counts bijective function from ${\Bbb Z}/p$ to ${\Bbb Z}/p$ and ${\Bbb Z}/p \times {\Bbb Z}/p$ acts on these by $f(x)\stackrel{(a,b)}{\rightarrow} f(x-a)+b$. Excluding functions
of the form $cx+d, c\not=0$, all orbits have size $p^2$. 



A particular combinatorial proof of Wilson's theorem
According to Dickson's History of the Theory of Numbers, this proof was first found by J. Petersen, Tidsskrift for Mathematik (3), 2, 1872, 64-5. (Petersen divides everything by 2, but the idea is the same.) Dickson gives a number of references to rediscoveries of Petersen's proof in the 19th century. Petersen was also apparently the first to discover the combinatorial proof of Fermat's theorem.


