تیۆرمی ویلسۆن

${\displaystyle \;(p-1)!\equiv -1{\pmod {p}}}$

${\displaystyle \prod _{k=1 \atop \gcd(k,m)=1}^{m}\!\!k\ \equiv {\begin{cases}-1{\pmod {m}}&{\text{if }}m=4,\;p^{\alpha },\;2p^{\alpha }\\\;\;\,1{\pmod {m}}&{\text{otherwise}}\end{cases}}}$ 

${\displaystyle (3-1)!=1\times 2\ =2\equiv -1{\pmod {3}}}$ 

${\displaystyle (5-1)!=1\times 2\times 3\times 4=24\equiv -1{\pmod {5}}}$ 

${\displaystyle (7-1)!=1\times 2\times 3\times 4\times 5\times 6=720\equiv -1{\pmod {7}}}$ 

${\displaystyle (11-1)!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10=3628800\equiv -1{\pmod {11}}}$ 

${\displaystyle (13-1)!=1\times 2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10\times 11\times 12=479001600\equiv -1{\pmod {13}}}$