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Math Help - True or False Statement?

  1. #1
    Junior Member
    Joined
    Aug 2007
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    32

    True or False Statement?

    If True explain show why and if false give a counterexameple.

    For All n>=1, if n is not prime, then neither is 2^n-1.

    How would I go about figuring this out? Is there a forumula or do I just plug numbers in?
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Paris
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    354
    Hi!

    A useful formula: \forall a,b,n\in\mathbb{N},\ n\geq 2,\ a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})


    Let n=pq be a positive integer with p,q\geq 2
    2^n-1=2^{pq}-1^q=(2^p)^q-1^q=(2^p-1)(\sum\limits_{k=0}^{q-1}2^k)

    Hence \forall n\geq 1\ \text{no}(n\ \text{prime})\Rightarrow\ \text{no}(2^n-1\ \text{prime})
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