\documentclass{amsart}

%% Use the format / typographical conventions of the AMS
%% (American Mathematical Society)
\usepackage{amsmath}

%% Allows clickable links to equations in references,
%% and also to web pages
\usepackage{hyperref}

%% Define new environments for theorems and similar
%% structures.  The typographical style 'plain' is
%% suitable for formal statements.
\theoremstyle{plain}

% The [section] parameter indicates that theorems are numbered
% according to the section they are in (Theorem 2.1 for instance)
\newtheorem{theorem}{Théorème}[section]

% The * indicates an unnumbered environment.
\newtheorem*{theorem*}{Theorem}

% the [theorem] parameter before the name of the environment
% indicates that the numbering of lemmas follows that of
% theorems: after Theorem 2.3, for instance, the next
% lemma will be Lemma 2.4
\newtheorem{lemma}[theorem]{Lemma}

% Similar definitions for propositions and corollaries
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

%% The 'remark' style gives environments where the main text is
%% in standard font (instead of italic, as far the 'plain' style).
\theoremstyle{remark}

% Since there is no indication before or after Remark, the
% remarks are numbered successively, independently of the
% numbering of the other items.
\newtheorem{remark}{Remark}

%% Indicate that equations should be numbered within each section
%% so there will be equation (1.1), then (1.2), etc, in Section 1,
%% and (2.1), etc, in Section 2.
\numberwithin{equation}{section}

%% Beginning of the actual document
\begin{document}

%% Give the title
\title{First example}

%% Command to actually typeset the title
\maketitle

%% Section
\section{First section}

%% Subsection
\subsection{Statements}

This is a formula:
%% A numbered equation, with a label for later reference
\begin{equation}\label{eq-gauss}
\int_{\infty}^{+\infty} e^{-\pi x^2}dx=1.
\end{equation}

%% One can align computations with the align and align*
%% environments. The lines are separated by \\, and the & symbol in
%% each line (which is not printed) indicates the point where the
%% lines are aligned: each letter after the & appears at the same
%% point horizontally
%% The align* version does not number the equations
\begin{align*}
    a&=b\\
    &=c.
\end{align*}

%% A Theorem; the [Wiles, 1994] is used for attribution, and is
%% typeset in a special way, depending on the theorem style. The label
%% allows easy later rerefences. The theorem will be numbered
%% automatically, and the system keeps track of the numbering in case
%% of changes, insertions, etc.
\begin{theorem}[Wiles, 1994]\label{th-wiles}
  If $n\geq 3$ and $x$, $y$, $z$ are integers such that $x^n+y^n=z^n$,
  then $xyz=0$.
\end{theorem}

%% A version where the theorem is not numbered.
\begin{theorem*}[Wiles, 1994]
  If $n\geq 3$ and $x$, $y$, $z$ are integers such that $x^n+y^n=z^n$,
  then $xyz=0$.
\end{theorem*}

%% A Lemma
\begin{lemma}
  If $p$ is a prime number congruent to $1$ modulo $4$, then $p$ is
  the sum of two squares of integers.
\end{lemma}

%% Another theorem
\begin{theorem}[Vinogradov, 1930]\label{th-vinogradov}
  Every large integer is the sum of three primes.
\end{theorem}

%% Subsection
\subsection{Proofs}

%% Using \ref to reference previous statements, or previous equations.
By the theorem of Wiles (Theorem~\ref{th-wiles}), we have...
By equation~(\ref{eq-gauss}), we have ...

%% A remark
\begin{remark}
  Remarkably, Theorem~\ref{th-wiles} is hard to prove.
\end{remark}

%% End of the document
\end{document}



