Mersenne Primes

A Mersenne prime is a prime number of the form  
\[M_n =2^n-1\]
  where  
\[n \gt 0\]
  is a positive integer.
\[2^n-1\]
  is not a prime number for every value of  
\[n\]
. In fact  
\[2^n-1\]
  is not a prime number for an even number greatyer than 2, since if  
\[n=2m\]
  where  
\[m\]
  is an integer, then  
\[2^n-2=2^{2m}-1=(2^m+1)(2^m-1)\]
  and this number is composite.
Even if  
\[n\]
  is odd, this does not giarantee a prime number for  
\[2^n-1\]
.
If  
\[n=11\]
- 11 is prime - but  
\[2^{11}-1=2047=89 \times 23\]
.
There are thought to be infinitely many Mersenne primes.