Modular Arithmetic/Wilson's Theorem

Wilson's theorem

A natural number $\alpha$ is a prime number, if and only if:

(\alpha -1)!\equiv -1{\pmod {\alpha }}

Where $\alpha !$ denotes the factorial of $\alpha$ , meaning the product of all numbers less than or equal to $\alpha$ , for every natural number.

Examples

5 is a prime number because,

4!=24

and

24\equiv -1{\pmod {5}}\implies 25|5

which is true. 6, on the other hand, is not, as

5!=120

and

120\equiv -1{\pmod {6}}\implies 121|6

which is false.