Mersenne Primes

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

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