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Math Help - Quadratic Gauss sums

  1. #1
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    Quadratic Gauss sums

    The quadratic Gauss sum G(m:n) is defined by

    G(m:n)={\displaystyle {\textstyle \underset{r=1}{\overset{n}{\sum}}\varpi^{mr^{2}}}}, where \varpi=e^{2\pi i/n}.

    a) Show that

    w^{mr^{2}}=\varpi^{ms^{2}} whenever mod\: n)" alt="r\equiv s\mod\: n)" />

    and deduce that G(m:n)=\Sigma\varpi^{mr^{2}}, where the summation extends over any complete set of residues.

    b) Let m and n be integers with (m,n)=1, and let r and s run through complete sets of residues mod m and mod n.

    Prove that t=nr+ms runs through a complete set of residues mod mn, and that

    t^{2}=n^{2}r^{2}+m^{2}s^{2}\;(mod\: mn).

    c) Use the results of parts (a) and (b) to prove that

    G(m:n)G(n:m)=G(1:mn), for (m,n)=1,

    and evaluate each term of this equation when m=3 and n=4.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    a.)  r\equiv s\bmod{n}\implies r=s+kn\implies r^2=s^2+2skn+k^2n^2 . Now what's  \varpi^{2skn} and  \varpi^{k^2n^2} ?

    Before I go any further, what are your ideas for the rest?
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  3. #3
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    Don't they both equal 1?

    To be honest, I'm not sure I really have any ideas for this question.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Yes.

    b.) Suppose  t=nr+ms and  t'=nr'+ms' where  t'\equiv t\bmod{mn}\implies t'=t+kmn .

    Now subtract the two equations and reach a contradiction.
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  5. #5
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    I'm being very stupid here....but what two equations?

    I'm also not sure how r^2=s^2 + 2 answers the first part of the question.
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  6. #6
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Cairo View Post
    I'm being very stupid here....but what two equations?

    I'm also not sure how r^2=s^2 + 2 answers the first part of the question.
     t=nr+ms and  t'=nr'+ms'

    And I don't know what you mean about your other question.
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  7. #7
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    In your first post, do you mean r=s-kn ?

    This then gives r^2 = s^2
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  8. #8
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Cairo View Post
    In your first post, do you mean r=s-kn ?

    This then gives r^2 = s^2
    That's the definition of mod.
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  9. #9
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    Thanks for your posts. The question is a tiny bit clearer, but I am still not convinced I fully understand how to answer the question. I'll keep working on it.
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