How does Wilson's theorem provide a test for prime numbers?

In number theory, Wilson’s theorem provides a definitive test for prime numbers based on the property that an integer $p>1$ is prime if and only if it divides the quantity $(p-1)!+1$.12

The theorem establishes this relationship in two ways:
* Primality: If $p$ is a prime number, then $(p-1)!+1$ is divisible by $p$.1
* Converse: If $n$ is a composite number, then $(n-1)!+1$ is not divisible by $n$.1

While this provides a mathematically absolute method to determine whether a number is prime, it is generally considered impractical for large numbers because calculating the factorial $(n-1)!$ becomes computationally exhaustive as the value of $n$ increases.1