Prove contrapositive of a statement after proving its converse. (If $n$ is prime then $2^n-1$ is also prime) I have been given the following problem:

Let n > 1 be a positive integer. Let P be the following statement:
If n is a prime number then $2^n - 1$ is a prime number
Write down the converse of the statement P. Is it always true? Justify your answer.

I have solved this part by using a proof by contradiction, in the same way that is shown in this question. However, this problem is followed up with the following:

Write down the contrapositive of the statement P.
Is it always true? Justify your answer.

From my understanding the contrapositive of P is:
If $2^n-1$ is not prime, then n is not prime
What I'm struggling with is how to go about proving this. Could I just use the same method as the previous part of the question (again, see here)? Or would I have to get the contrapositive of the contrapositive (resulting back in P) and prove that in some way?
 A: I think you mean converse. We can use a proof by exhaustion to prove the converse and disprove the contrapositive. As @lulu pointed out in the comments, $2^{11}-1=2047$, which is composite, but $11$ is still prime, which means that $2^n-1$ does not need to be prime for $n$ to be prime. I found another example as well:  $n=23$. I hope this answers your question. Don't hesitate to ask any questions in the comment section below.
A: I just want to make another point. Note that, there is no simple computable and useful algebraic expression or function $f(n)$, such that if $n$ is prime, then $f(n)$ is also a prime number. Unfortunately, such a simple formula currently doesn't exist, as far as I know.
A: The converse statement to statement (A) below:

(A) If $n$ is a prime number, then $2^n-1$ is a prime number.

is statement (B):

(B) If $2^n-1$ is a prime number, then $n$ is a prime number.

Likwise, statement (A) is the converse to statement (B).
Statement (A) is false but statement (B) is true.
To see that (A) is false, note that $2047=2^{11}-1$ as per lulu's comment, and $11$ is prime but $2047$ is infact not. So (A) [and thus the contrapositive to (A)] is not true.
On the other hand, (B) is true. We show this by working with the contrapositive of (B), which we will write as (B$'$):

(B$'$) If $n$ is not prime, then $2^n-1$ is not prime.

To establish (B$'$), first suppose that $n$ is not prime. Then write $n=pq$ where both $p$ and $q$ are integers at least $2$. Then $$2^n-1=2^{pq}-1$$ $$=$$ $$(1+2^q+2^{2q}+\ldots +2^{(p-1)q})(2^p-1),$$ and both
$1+2^q+\ldots + 2^{(p-1)q}$ and $2^p-1$ are at least $2$ [actually at least $3$], because $p$ and $q$ are both integers at least $2$, and so $2^p$, $2^{(p-1)q}$ are each at least $2^2=4$. So $2^n-1$ is a product of $2$ integers each at least $2$, so $n$ is not prime. Thus (B$'$) is true.
As (B) is precisely the contrapositive of (B$'$) and (B$'$) is true, it follows that (B) must be true as well.
