# give an example

• Feb 16th 2010, 08:54 AM
flower3
give an example
Prove that if $\displaystyle a^n -1$ is prime where $\displaystyle a \in N$ and $\displaystyle n \geq 2,$ then $\displaystyle a =2$ and $\displaystyle n$ is prime
How I can use the identity : $\displaystyle a^{kl} -1$=$\displaystyle (a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + …+$\displaystyle a^{k }+1)$ to solve this Q ??
• Feb 16th 2010, 09:44 AM
tonio
Quote:

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

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

UNLESS $\displaystyle a=2$ ...(Rock)

Tonio