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\title{Elementary Number Theory II, final examination}
%\author{David Pierce}
\date{June 2, 2008}
\usepackage{verbatim}  % allows a comment environment:
%\begin{comment}
\address{Mathematics Dept\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}
%\end{comment}

\usepackage{amsmath,amsthm,amssymb}
\usepackage{url}
\usepackage{textcomp}  % supposedly useful with \oldstylenums
\usepackage{hfoldsty} % this didn't work until I added missing
		      % brackets to some of the files.

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\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}
%\renewcommand{\theenumi}{\roman{enumi}}
%\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}

\renewcommand{\theenumii}{\roman{enumii}}
\renewcommand{\labelenumii}{\textnormal{(\theenumii)}}

%%%%%%%%%%%%%%%
\newcommand{\included}{\subseteq}      % [the name suggests the meaning here]

\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\Q}{\stnd{Q}}         % rationals

\newcommand{\mi}{\mathrm i}
%\newcommand{\gr}{\upphi}  % golden ratio
\newcommand{\mLambda}{\mathit{\Lambda}}
\newcommand{\mPi}{\mathit{\Pi}}
\newcommand{\mMu}{M}

\newcommand{\norm}[1]{\operatorname{N}(#1)}

\newcommand{\size}[1]{\lvert#1\rvert}


\let\oldsqrt\sqrt
\renewcommand{\sqrt}[2][1]{\oldsqrt{\vphantom{#1}}#2}
%\newcommand{\rft}{\sqrt{14}}
%\newcommand{\rtt}{\sqrt{13}}

\newcommand{\lat}[1][\alpha,\beta]{\langle#1\rangle} % lattice
\newcommand{\ord}[1][\mLambda]{\mathfrak O_{#1}}
\newcommand{\roi}{\ord[K]}

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\newtheorem{problem}{Problem}

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%{\begin{proof}[Solution.]}%
%{\end{proof}}

%\numberwithin{equation}{section}
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\begin{document}
  \maketitle\thispagestyle{empty}

\begin{comment}
My solutions to the following will be at
\begin{center}
  \url{http://www.math.metu.edu.tr/~dpierce/3x122final-sol.pdf}\\
\url{http://www.math.metu.edu.tr/~dpierce/3x122final-sol.ps}
\end{center}
\end{comment}


  \begin{problem}
    Find the positive rational-integer solutions to
    \begin{equation*}
      x^2-22y^2=3.
    \end{equation*}
  \end{problem}
\vfill
  \begin{problem}
    The curve $E$ defined by the cubic equation
    \begin{equation*}
      y^2=x^3-2
    \end{equation*}
has the rational point $(3,5)$.  This problem is about obtaining other
rational points.
\begin{enumerate}
  \item
Find an equation for the tangent line to $E$ at $(3,5)$.  (You may use
implicit differentiation.)
\item
This tangent line meets $E$ twice at $(3,5)$.  Find the third point of
intersection.  (You may use that the sum of the roots of
$x^3-Ax^2+Bx-C$ is $A$.)
\item
Now generalize:  Suppose $(a,b)$ is on $E$, and let $\lambda$ be the
slope of the tangent line to $E$ at $(a,b)$.  Find $\lambda$ (assuming
$b\neq0$). 
\item
Derive the conclusion that this tangent line meets $E$ also at
\begin{equation*}
  \Bigl(\frac{a^4+16a}{4b^2},\frac{-a^6+40a^3+32}{8b^3}\Bigr).
\end{equation*}
\end{enumerate}
  \end{problem}
\vfill
\end{document}

