# Math Help - give an example

1. ## give an example

Prove that if $a^n -1$ is prime where $a \in N$ and $n \geq 2,$ then $a =2$ and $n$ is prime
How I can use the identity : $a^{kl} -1$= $(a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + …+ $a^{k }+1)$ to solve this Q ??

2. Originally Posted by flower3
Prove that if $a^n -1$ is prime where $a \in N$ and $n \geq 2,$ then $a =2$ and $n$ is prime
How I can use the identity : $a^{kl} -1$= $(a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + …+ $a^{k }+1)$ to solve this Q ??

Well, if $n=kl$ is not a prime then, as you wrote, you have $a^{kl} -1= (a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + …+ $a^{k }+1)$ , and this is a non-trivial factorization of $a^n-1$

UNLESS $a=2$ ...

Tonio