## 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.