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Math Help - Mersenne Prime Question

  1. #1
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    Mersenne Prime Question

    Let p be an odd prime of the form p = 4k + 3

    a) Prove that if q = 2p + 1 is prime, then q divides the Mersenne number M_{p} = 2^p - 1.

    b) Use part (a) to prove that M_{11}, M_{23}, and M_{251} are composite numbers.
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  2. #2
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    (b) should be easy enough since I know that everything from (a) must hold and I have specific examples.

    I don't really get what I'm supposed to use to prove (a). What does q being prime have to do with it dividing the Mersenne number?
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    a.

    If p=4k+3 then q=8k+7

    q|M_n , in other words is:

    2^{(q-1)/2}=2^p \equiv 1(mod q)

    It is actually the same to say that Legendre's symbol (2/q)=1.

    {(2/q)=1, when q=1(mod 8) or q=7(mod 8)}
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