Is it true that in $2^n-1$, when $n$ is a prime number, you don't always get a Mersenne prime? For $2^n-1$, where $n$ is a prime number, is it true that you don't always get a Mersenne prime?  Remember, a Mersenne prime is a number that has a power of two subtracted by one and is then prime:$$n|2^n-1$$$$2|2^2-1$$$$=3$$$$3|2^3-1$$$$=7$$$$5|2^5-1$$$$=31$$$$7|2^7-1$$$$=127$$$$11|2^{11}-1$$$$=2,047$$
Well, $2,047=23\times89$
So, 11 can't be used for $n$, even though it's a prime number.  How does this happen that $n$ can't always be a prime number to fit into $2^n-1$ to give out a Mersenne prime?

This is a bit challenging to answer. In short, there is no reason why $2^p - 1$ would always give a prime number. You have just given evidence to this fact.
A Mersenne prime is a prime of the form $2^p - 1$. This definition does not mean that every number of the form $2^p - 1$ is prime.
