%\documentclass[a4paper,twoside,draft]{article}
\documentclass[a4paper,twoside,draft]{amsart}

\title{Number-theory exercises, III}
\author{David Pierce}
\date{\today}

\address{Mathematics Dept\\
Middle East Technical University\\
Ankara 06531, Turkey}

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


\usepackage{amssymb,amsmath,amsthm}
%\usepackage{amscd}     % commutative diagram
%\usepackage[mathscr]{euscript}
\usepackage{url}
\usepackage{verbatim}  % allows a comment environment:

%\usepackage{parskip}   % paragraphs not indented, but separated by spaces

\usepackage{eco}  % gives old-style numerals

\input{abbrevs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Theorem-like environments  %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\swapnumbers


%\newtheorem{theorem}{Theorem}
%\newtheorem{axdef}[theorem]{Axiom and definition}
%\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}

%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{parag}[theorem]{}
%\newtheorem{ex}[theorem]{Exercise}
\newtheorem{ex}{Exercise}

\theoremstyle{remark}

%\newtheorem{remarks}[theorem]{Remarks}
%\newtheorem{remark}[theorem]{Remark}

%\newcommand{\rmk}[1]{\marginpar[\flushright#1]{\flushleft#1}}

\begin{document}
  \maketitle\thispagestyle{empty}

Here $p$ and $p_i$ are always prime numbers.

  \begin{ex}
    $p\equiv\pm1\pmod6$.
  \end{ex}

  \begin{ex}
    If $p\equiv1\pmod3$ then $p\equiv1\pmod6$.
  \end{ex}

  \begin{ex}
    If $n\equiv2\pmod3$, then $n$ has a factor $p$ such that
    $p\equiv2\pmod3$. 
  \end{ex}

  \begin{ex}
    Find all primes of the form $n^3-1$.
  \end{ex}

  \begin{ex}
    Find all $p$ such that $3p+1$ is square.
  \end{ex}

  \begin{ex}
    Find all $p$ such that $p^2+2$ is prime.
  \end{ex}

  \begin{ex}
    $n^4+4$ is composite.
  \end{ex}

  \begin{ex}
    If $n$ is positive, then $8^n+1$ is composite.
  \end{ex}

  \begin{ex}
    Find all integers $n$ such that the equation
    \begin{equation*}
      x^2=ny^2
    \end{equation*}
has only the zero solution.  Prove your findings.
  \end{ex}

  \begin{ex}
    If $p_0<\dotsb<p_n$, prove that the sum
    \begin{equation*}
      \frac1{p_0}+\dotsb+\frac1{p_n}
    \end{equation*}
is not an integer.
  \end{ex}

%\bibliographystyle{plain}
%\bibliography{../../../TeX/references}

%\layout


\end{document}

