Factoring and Using the Formula

An interesting formula is `x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)`.

A: Use it to factor the expression `a^n-b^n`.

B: Factor the expression `a^n+b^n` for any odd integer n.

C: Prove that if `2^n-1` is prime, then n is also prime.

D: Prove that if `2^n+1` is prime, then n is necessarily a power of 2, which is equivalent to `n=2^m`