TITLE: Disprove that "if $p$ is a prime number, then $2^p-1$ is also a prime number"?
QUESTION [9 upvotes]: We can see manually that $2^p-1$ is not prime. As $2047$ is not a prime. $2^{11} = 2048$.
But I'm unable to figure out a formal way of disproving the statement.

REPLY [3 votes]: You could also do a proof by contradiction (which in this case would only be a more formal way to phrase your proof).  
In this method you start by assuming that the opposite of what you want to prove is true (in our case, assuming that $2^p - 1$ is a prime). Given what we know, 2047 should be prime, because it follows the form of $2^p - 1$.  
However, we can see that:  

$2^{11} - 1 = 2048 - 1 = 2047$  

We know 2047 is not a prime number, because 2047 = 23 * 89.  
Now we have reached the contradiction part of the proof. According to our assumption in the beginning of the proof, any number of the form $2^p - 1$ should be prime. We have just shown this assumption to lead to a contradiction, and thus we must abandon this assumption as false.