Definition of wilson's theorem

Question

The statement of wilson's theorem states that, for any prime number n>1

$( n-1 )! \equiv -1 \mod n$

Is this equivalent to saying that

$( n-1 )! \equiv n-1\mod n$

If no, what is the difference ?, If yes, why are we doing the same thing in $2$ different ways?

Thanks

Answer 1

These two formulations are the same because $n-1 \equiv -1 \pmod{n}$. That is independent of Wilson's theorem.

Which formulation you prefer is a matter of taste. The one with $n-1$ is the least positive remainder.