I have the following question where I need to provide a counterexample
if $n$ is prime, then $2^n -1$ is prime
The statement is invalid and I now that $2^n -1$ is not prime when $ n = 11$, but other than plugging in prime numbers is there any other way to figure out whether $2^n -1$ is prime if $n$ is prime?
No. There's a way to test for prime $n$ whether $2^n-1$ is prime without computing it first, but showing some $n$ fails that test still requires manually finding a counterexample. A proof that didn't have this requirement would be nonconstructive; but I doubt there's a nonconstructive proof not all prime $n$ obtain prime $2^n-1$.