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 $\displaystyle n=4$.