If I square a prime number ( >3) and subtract 1 , it turns out it is a multiple of 24 and there are different ways of proving this.

Why doesn't my approach below work?

All prime number are of the form 6n+1 or 6n-1 ( greater than 3).

So (6n+1)^2 -1 or (6n-1)^2 -1 only proves that they are multiples of 12 not 24?

What am I missing?