% LESSON THREE
\documentclass{amsart}
\thispagestyle{empty}
\newtheorem{thm}{Theorem} % This is called a 'proclamation.'
% The first argument is the environment name. The second is what will be typeset.
% You can also use this for lemmas, corollaries, definitions, and so on.
% The theorems will be numbered consecutively: 1, 2, 3, ...
\theoremstyle{definition}
% This changes the style from 'plain' (the default) for the next proclamation.
\newtheorem*{dfn}{Definition}
% As in the align environment, the asterisk here suppresses the numbering.
\theoremstyle{remark}
% And this changes the style to 'remark', the third and final option.
\newtheorem*{note}{Note}
\begin{document}
\begin{center}
\textbf{Lesson Three: Proclamations and Proofs}
\end{center}
\begin{dfn}
An integer $m$ is \emph{even} if $m=2j$ where $j$ is an integer.
% Note the effect of the \emph style in this situation.
\end{dfn}
\begin{thm}
The sum of any two even integers is an even integer.
\end{thm}
\begin{proof}
Suppose $m$ and $n$ are even integers.
By the definition of even, $m=2j$ and $n=2k$ where $j$ and $k$ are integers.
Therefore, $m+n=2j+2k=2(j+k)$.
Since the integers are closed under addition, $j+k$ is also an integer.
So, $m+n$ is twice another integer ($j+k$), meaning $m+n$ is even, as desired.
\end{proof}
\begin{thm}
The equation $x^n+y^n=z^n$ has no non-zero integer solutions for $n>2$.
\end{thm}
\begin{proof}
I have a marvellous proof of this, but the page is too small to contain it.
\end{proof}
\begin{thm}
The number 8675309 is prime.
\end{thm}
\begin{proof}
Just ask Jenny.
\end{proof}
\begin{dfn}
A \emph{mathematician} is a device for turning coffee into theorems.
\emph{[attributed to Paul Erd\H{o}s]}
\end{dfn}
\begin{note}
Begin laughing now.
\end{note}
\end{document}