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 $\displaystyle n=4$.

2. 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?

3. Don't they both equal 1?

To be honest, I'm not sure I really have any ideas for this question.

4. 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.

5. 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.

6. Originally Posted by Cairo
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 $\displaystyle t'=nr'+ms'$

7. In your first post, do you mean r=s-kn ?

This then gives r^2 = s^2

8. Originally Posted by Cairo
In your first post, do you mean r=s-kn ?

This then gives r^2 = s^2
That's the definition of mod.

9. 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.