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Math Help - Number theory question 2-3

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    Number theory question 2-3

    Show that if 2^{p} -1 is prime, then p must be prime
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  2. #2
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    Re: Number theory question 2-3

    Quote Originally Posted by maximus101 View Post
    Show that if 2^{p} -1 is prime, then p must be prime
    I recommend proving the "contrapositive". Assume p is NOT prime and show that 2^p- 1 is not prime.

    It will help to use x^n- 1= (x- 1)(x^{n-1}+ x^{n-2}+ x^{n-3}+ \cdot\cdot\cdot+ x+ 1). You can't apply that to 2^p- 1 directly because, taking x= 2, x- 1= 2- 1= 1 so that's not a new factor.
    Last edited by HallsofIvy; October 18th 2012 at 02:09 PM.
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    Re: Number theory question 2-3

    Another related question, prove that 2^2k - 1 is divisible by 3 always.

    Salahuddin
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