\documentclass{amsart}

% the following set-up ensures that theorems etc are numbered consecutively
% within each section as 3.1, 3.2, etc.  
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]  
\newtheorem{lemma}[theorem]{Lemma} 
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\theoremstyle{remark}
\newtheorem*{example}{Example}

\numberwithin{equation}{section}

\begin{document}
\title{Writing a mathematical research report}
\author{MAT 3166 student}
\date{\today}
\begin{abstract}
The abstract is a very short summary of the results in the
work.  It should be more illuminating than the title, but
not as detailed as the introduction.
\end{abstract}
\maketitle

\section{Introduction} 

Give a brief statement of the problem, including its
historical context and an overview of the major developments
in its history, with references~\cite{Nev}.

Provide a guide for the reader.  In Section~\ref{S:background}
we'll discuss some particular conventions for the writing of
mathematical articles.  
In Section~\ref{S:problem} we'll define the problem
precisely    and give a proof of the main result.  We conclude
in Section~\ref{S:applications} with some examples and
applications.

\section{Mathematical Conventions} \label{S:background}

Most mathematical journals rely on authors to electronically
submit their articles in \LaTeX.  The journals usually provide
a \emph{style file} which allows an author to easily recreate
the ``look and feel'' of the particular journal.  For this report,
the default \texttt{amsart} documentclass will be fine.

We write math papers in first person plural, signifying ``the
author(s) and the reader together''.  Besides being a more natural
writing style than the stiff formality of ``one sees that'' 
or ``in this paper it is shown that'', it tends to be a bit
shorter.

Math papers don't have a conclusion, even though articles in most
other disciplines do.  Instead, the introduction takes up the slack.
So most articles will include in the introduction:
\begin{itemize}
\item a discussion
of the context of the work;
\item a statement of 
the main theorem or theorems;
\item a discussion of related results in the literature; and
\item a ``roadmap'' of the paper, detailing what will be found in
each section.
\end{itemize}
For a paper whose goal is to present a new theorem, for example, 
the main body of the paper will have minimal discussion, in general,
focussing instead on presenting the results and giving the proofs.
Where the introduction may be packed with several disjoint things,
the rest of the paper will flow smoothly.

When a new term is introduced, use \emph{italics} (command: emph),
not boldface or underlining.  \LaTeX has several features to help
typeset definitions or theorems consistently:  \texttt{theoremstyle}
commands.  (See the preamble of this document for one set-up.)

\begin{theorem} \label{T:t1}
Footnotes are rarely used.  References to the literature are made
as they occur~\cite{Gauss}.  They should be made as specific as
possible~\cite[Thm3.3]{Gauss} if the work is a long one.
\end{theorem}

\begin{proof}
The bibliography should be typeset and arranged as uniformly as
possible.  Many different ways of citing a particular work may be
acceptable; the key points are to identify:  the author, the
title of the work, the publisher, and the year.  More detail
is usually needed for articles in journals, such as volume or issue
number.  Book and journal titles may be put in \emph{italics}, whereas 
article titles are ``put in quotation marks''.
\end{proof}

Cross-referencing within your article can be very helpful to the
reader.  For example, refer to Theorem~\ref{T:t1} or Equation~(\ref{E:gauss}).

\section{Statement of the Problem} \label{S:problem}

The most important problem in mathematics involves 
simplifying the expression
$$
\sum_{i=1}^n i,
$$
a problem famously solved by Gauss~\cite{Gauss} at the age of six.
His solution
\begin{equation} \label{E:gauss}
\sum_{i=1}^n i = \frac{n(n+1)}{2}
\end{equation}
is brilliant.

Next, we'd like to understand the $n$th term in a 
sequence $x_0,x_1,x_2,\ldots$, where $x_i = 4i+2$.

\section{Examples} \label{S:applications}

There are many wonderful applications of number theory.

\begin{thebibliography}{999}
\bibitem[B]{Nev} R.~U.~Brilliant, ``The Quadratic Form'' in \emph{Journal of
Applied Number Theory}, Vol. 3, No. 2 (1998).
\bibitem[G]{Gauss} C.~F.~Gauss, \emph{Brilliant Math}, Princeton University Press, 2002.
\end{thebibliography}

\end{document}
