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Math Help - x^7 = 12 mod 29 (primitive roots)

  1. #1
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    x^7 = 12 mod 29 (primitive roots)

    I need to solve the following problem:
    "=" means is congruent to.
    x^7 = 12 mod 29

    This is what I have:
    I am using q=2 as the primitive root mod 29.
    7*I(x)=I(12) mod 28
    7*I(x)=7 mod 28
    I(x) = 1 mod 4

    What do I do next?
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  2. #2
    Junior Member
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    never mind, I got it. i will post the solution later
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Well  (x,29)=1 , so we know  x\equiv2^y\bmod{29} .

    Therefore  \text{ind}_2(2^y) = y\equiv1\bmod{4}

    Our solutions are thus  x=\{2^1,2^5,2^9,2^{13},2^{17},2^{21},2^{25}\} .

    Note that these are the only solutions since  2 is a primitive root, so  \{2^0,\ldots,2^{28}\} generate all possibe solutions.
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