Mersenne Primes
\[M_n =2^p-1\]
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.
