But why do we have to "discover" it when we know the formula would be 2^N - 1...? Are we trying to prove a corollary or what?
Not all 2^N - 1 are prime. For example, N=18 makes 2^N - 1 = 262143, which can also be written as 3^3 * 7 * 19 * 73 (not prime).
A lot simpler example is 2^4-1=5*3