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: $\displaystyle 2^{n-1}$ is prime for only one value of $\displaystyle n$.

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 $\displaystyle 2^n-1$, so that the question is less trivial...

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