Proof that $2^n-1$ does not always generate primes when primes are plugged in for $n$?
Solution 1:
$$\Large 2^{11}-1=23\cdot 89$$ Take a look at the Wikipedia page on Mersenne primes. There are (currently) only $48$ known prime numbers $p$ such that $2^p-1$ is prime, after people have used computers to check millions.