Hi,
How to show that $\displaystyle (n-1)!\equiv-1[n] \Longrightarrow n \in \mathbb{P}$????
I take it $\displaystyle \mathbb{P}$ are the prime numbers? This is Wilsons theorem.
For any $\displaystyle a \in \{1, 2, \dots, p-1\}$ there's an $\displaystyle a' \in \{1, 2, \dots, p-1\}$ with $\displaystyle aa' \equiv 1$ (mod p). Further, $\displaystyle a' = a$ iff $\displaystyle a = 1$ or $\displaystyle p-1$. Hence, in $\displaystyle (p-1)!$, the numbers forming this product can be cancelled in pairs, apart from $\displaystyle 1$ and $\displaystyle p-1$. Hence result.