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Math Help - Pairs of primes and Cardinalities of sets

  1. #1
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    Pairs of primes and Cardinalities of sets

    For a pair of primes (p_0,p_1), let Condition A be the following:
    For all integers n>0, \mbox{card}\left( \left\{ p_{1-j}^i \in \mathbb{Z} / p_j^n \mathbb{Z} \mid i \in \mathbb{N} \right\} \right) = p_j^n-p_j^{n-1} for both j=0 and j=1.

    Let M be the set of all pairs of primes that satisfy Condition A. I can show that (2,3) \in M, so M is nonempty. I think it might be true that (p_0,p_1) \in M if p_0<p_1 and there does not exist a prime p_2 such that p_0<p_2<p_1.

    Let K = \left\{(p_0,p_1) \mid p_0,p_1 \mbox{ prime}, p_0<p_1, \nexists p_2 \mbox{ prime s.t. }p_0<p_2<p_1\right\}. Then I think it may be true that M = \left\{(p_0,p_1) \mid (p_0,p_1) \in K \mbox{ or } (p_1,p_0) \in K\right\}.

    Is there an obvious counterexample? (So far, I haven't found any, but I admit I haven't looked terribly hard yet).
    Last edited by SlipEternal; September 26th 2013 at 06:47 AM.
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  2. #2
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    Re: Pairs of primes and Cardinalities of sets

    Ok, I figured this out. I am essentially looking for pairs of primes p,q for which p is a primitive root modulo q^n for all n>0 and q is a primitive root modulo p^n for all n>0. I realized that (2,3)\notin M, I have no clue what I was thinking. And it is possible that M is empty. My bad. Thanks anyway.
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