Question

if $2^n-1$ a prime, show that $n$ is a prime.

Answer 1

Suppose that $n$ is composite,

hence can be written $n=ab$ with $a$ and $b>1$. Then $2^n-1=2^{ab}-1$

$=(2^a{)}^b-1$

$=(2^a-1)\left(\right(2^a{)}^{b-1}+(2^a{)}^{b-2}+.......+2^a+1)$

So $2^n-1$ is composite. By contropositive, If $2^n-1$ is prime then $n$ is prime.