## 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.