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Math Help - prime

  1. #1
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    prime

    Hi

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

    thank you
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  2. #2
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    Quote Originally Posted by qwerty321 View Post
    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.
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  3. #3
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    erm 2^1 is the only power of 2 which is prime which mean n = 2...
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  4. #4
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    Quote Originally Posted by qwerty321 View Post
    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.
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