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Thread: False witnesses

  1. #1
    Dec 2009

    False witnesses

    I'm stuck on one part of a problem.

    "suppose $\displaystyle \Phi(n)=2t$ where t is odd, and $\displaystyle n=p^e$ for odd prime p and positive integer e.
    Let L be the subgroup of false witnesses, i.e. L is the set of congruence classes made up of integers b prime to n such that $\displaystyle b^{(n-1)/2)}\equiv \left(\frac{b}{n}\right)\ mod n$

    Now suppose the order of L is $\displaystyle |L|=\Phi(n)/2=t$.

    Let g be a primitive root mod n,so g has order $\displaystyle \Phi(n)$ and every element prime to n is of the form $\displaystyle g^j mod n$ for some j.

    Show that for every integer $\displaystyle g^j$ belonging to a congruence class in L, j must be even. Since |L|=t, explain why $\displaystyle L=g^{2i}: i=1....t$"

    I really don't know what to do here so any help would be appreciated.
    Last edited by xRoad; Dec 21st 2009 at 08:20 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
    Jun 2009
    Are you sure you posted the problem exactly?
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  3. #3
    Dec 2009
    Sorry, the question is a little confusing, but everything I typed is correct, but I did leave a part out at the beginning,

    "$\displaystyle n=p^e$, suppose $\displaystyle p-1=2s$ where s is an odd integer. Show that $\displaystyle \phi(n)=2t$, where t is odd."

    I think I did that part correctly. The bulk of the problem is trying to show that

    Suppose it happens that $\displaystyle |L|=\phi(n)/2=t$

    is impossible through 3 steps, so it's kind of a guided proof by contradiction (or so i think).

    1. The first step is to explain why every element of L must have odd order. I think I did this part correctly but I'm not 100% positive.

    2. The next part is the question about the primitive g mod n.

    3. Show that $\displaystyle g^{n-1} \equiv 1 mod n $, and use that to show $\displaystyle \phi(n)|(n-1)$. Why is this impossible.

    So I thought the actual contradiction comes from part three, which I haven't even gotten to yet.
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