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Name (Last, First): $\underline{\hspace{3in}}$ \hfill Student ID: $\underline{\hspace{2in}}$

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{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1516}{2015-16}:
  \href{\myurl/classes/16-475-ProblemSolving}{MAT 475 Problem Solving Seminar}:
  \hfill\url{http://drorbn.net/16-475}
}

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{\large\bf Quiz 4} on February 4, 2016: ``Modify the Problem'' / ``Choose an Effective Notation''. You have 30 minutes to solve as much as you can of the problems below. Please write on both sides of the page. \hfill {\bf Good Luck!}

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\noindent{\bf Problem 1} (Larson's 1.4.3). Prove that there does not exist positive integers $x$, $y$, and $z$ such that $x^2+y^2+z^2=2xyz$.

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\noindent{\bf Problem 2} (Larson's 1.5.10).  A well known theorem asserts that a
prime $p>2$ can be written as a sum of two perfect squares ($p=m^2+n^2$
with $m$ and $n$ integers) iff $p$ is 1 mod 4. Assuming this, prove:
\begin{enumerate}
\item Every prime which is 1 mod 8 can be written as $x^2+16y^2$, with $x$
and $y$ integers.
\item Every prime which is 5 mod 8 can be written as $(2x+y)^2+4y^2$, with
$x$ and $y$ integers.
\end{enumerate}

\noindent{\bf Problem 3} (no credit, yet the best solutions will be advertised). What is your favourite ``Modify the Problem'' or ``Choose an Effective Notation'' problem?

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