\[M_n =2^p-1\]

where \[p \gt 1\]

is a positive prime number.\[2^n-1\]

is not a prime number for every value of \[n\]

. In fact \[2^n-1\]

is not a prime number for an even number greater 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 or an odd prime, this does not guarantee 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.