# Math Help - prime

1. ## prime

Hi

how can i show that if 2^(n-1) is prime then n is prime?

thank you

2. Originally Posted by qwerty321
how can i show that if 2^(n-1) is prime then n is prime?
HINT: $2^{n-1}$ is prime for only one value of $n$.

3. erm 2^1 is the only power of 2 which is prime which mean n = 2...

4. Originally Posted by qwerty321
Hi

how can i show that if 2^(n-1) is prime then n is prime?

thank you
I guess you meant $2^n-1$, so that the question is less trivial...

Let $d\geq 0$ be a diviser of $n$. Write $n=d\times q$. Apply the formula $a^q-1=(a-1)(1+a+\cdots+a^{q-1})$ to $a=2^d$ to conclude that $2^d-1$ divides $2^n-1$. Since $2^n-1$ is prime, this means $2^d-1=0$ or $2^d-1=2^n-1$, hence $d=0$ ou $d=n$. This proves that $n$ is prime.