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Thread: Quadratic Gauss sums

  1. #1
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    Quadratic Gauss sums

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

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

    a) Show that

    $\displaystyle w^{mr^{2}}=\varpi^{ms^{2}}$ whenever $\displaystyle r\equiv s\mod\: n)$

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

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

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

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

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

    $\displaystyle G(m:n)G(n:m)=G(1:mn)$, for $\displaystyle (m,n)=1$,

    and evaluate each term of this equation when $\displaystyle m=3$ and $\displaystyle n=4$.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    a.) $\displaystyle r\equiv s\bmod{n}\implies r=s+kn\implies r^2=s^2+2skn+k^2n^2 $. Now what's $\displaystyle \varpi^{2skn} $ and $\displaystyle \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 $\displaystyle t=nr+ms $ and $\displaystyle t'=nr'+ms' $ where $\displaystyle 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.
    $\displaystyle t=nr+ms $ and $\displaystyle 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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