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\noindent \textbf{Math 465 - Fall 2019 \\
 Homework 10} \\
Due December 11, 2019 \vspace{-4mm}\\

\noindent \small \textit{Mathematics is the queen of sciences and number theory is the queen of mathematics.
}\\\vspace{1mm} \hfill--- Carl Gauss

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\begin{enumerate}

\item Problem 11-1.1.

\item By imitating our proof in class (using Thue's Theorem) and the symbol $\left(\frac{-2}{p}\right)$, show that any prime $p \equiv 1,3 \pmod{8}$ can be written as $$x^2 + 2y^2 = p$$ in integers, $x$ and $y$.

\item Show that \[ \lim_{N\to \infty} \frac{\sum_{n=0}^N r_3(n)}{N^{3/2}} = \frac{4\pi}{3} \]
where $r_3(n)$ counts the number of ways to represent $N=x^2+y^2+z^2$.  Hint: follow the method of proof of the result for $r_2(n)$, using a sphere and unit cubes.



\end{enumerate}

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