Andrew Tulloch | LaTeX2Markdown Demonstration

# Andrew Tulloch

## LaTeX2Markdown Demonstration

The following document was generated entirely by the LaTeX2Markdown tool. See below for the generated Markdown and the source LaTeX. The LaTeX source, Markdown source, and PDF (generated from the LaTeX document) are also available.

## Usage

### Simple Examples

This section introduces the usage of the LaTeX2Markdown tool, showing an example of the various environments available.

#### Theorem 1 (Euclid, 300 BC)

There are infinitely many primes.

#### Proof

Suppose that $p_1 < p_2 < \dots < p_n$ are all of the primes. Let $P = 1 + \prod_{i=1}^n p_i$ and let $p$ be a prime dividing $P$.

Then $p$ can not be any of $p_i$, for otherwise $p$ would divide the difference $P - \left(\prod_{i=1}^n p_i \right) - 1$, which is impossible. So this prime $p$ is still another prime, and $p_1, p_2, \dots p_n$ cannot be all of the primes.

#### Exercise 1

Give an alternative proof that there are an infinite number of prime numbers.

To solve this exercise, we first introduce the following lemma.

#### Lemma 1

The Fermat numbers $F_n = 2^{2^{n}} + 1$ are pairwise relatively prime.

#### Proof

It is easy to show by induction that [ F_m - 2 = F_0 F_1 \dots F_{m-1}. ] This means that if $d$ divides both $F_n$ and $F_m$ (with $n < m$), then $d$ also divides $F_m - 2$. Hence, $d$ divides 2. But every Fermat number is odd, so $d$ is necessarily one. This proves the lemma.

We can now provide a solution to the exercise.

#### Theorem 2 (Goldbach, 1750)

There are infinitely many prime numbers.

#### Proof

Choose a prime divisor $p_n$ of each Fermat number $F_n$. By the lemma we know these primes are all distinct, showing there are infinitely many primes.

### Available environments

We can format italic text, bold text, and code blocks.

1. A numbered list item
2. Another numbered list item
• A bulleted list item
• Another bulleted list item

#### Theorem 3

This is a theorem. It contains an align block. All math environments supported by MathJaX should work with LaTeX - a full list is available on the MathJaX homepage.

Maxwell’s equations, differential form. \begin{align} \nabla \cdot \mathbf{E} &= \frac {\rho} {\varepsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}} {\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \\ \end{align}

#### Theorem 4 (Theorem name)

This is a named theorem.

This is a lemma.

#### Proposition 1

This is a proposition

#### Proof

This is a proof.

This is a code listing.

### LaTeX Code

The following is the entire LaTeX document used to generate this post.

\documentclass[12pt]{amsart}
\usepackage{amsthm, amsmath, amssymb}
\usepackage{setspace}
\usepackage{listings}
\onehalfspacing

\theoremstyle{plain}% default
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{exer}[thm]{Exercise}

\title{LaTeX2Markdown Examples}
\author{Andrew Tulloch}
\begin{document}

% LaTeX2Markdown IGNORE
\maketitle
% LaTeX2Markdown END

\section{Simple Examples}

This section introduces the usage of the LaTeX2Markdown tool, showing an example of the various environments available.

\begin{thm}[Euclid, 300 BC]
There are infinitely many primes.
\end{thm}

\begin{proof}
Suppose that $p_1 < p_2 < \dots < p_n$ are all of the primes. Let $P = 1 + \prod_{i=1}^n p_i$ and let $p$ be a prime dividing $P$.

Then $p$ can not be any of $p_i$, for otherwise $p$ would divide the difference $P - \left(\prod_{i=1}^n p_i \right) - 1$, which is impossible. So this prime $p$ is still another prime, and $p_1, p_2, \dots p_n$ cannot be all of the primes.
\end{proof}

\begin{exer}
Give an alternative proof that there are an infinite number of prime numbers.
\end{exer}

To solve this exercise, we first introduce the following lemma.
\begin{lem}
The Fermat numbers $F_n = 2^{2^{n}} + 1$ are pairwise relatively prime.
\end{lem}

\begin{proof}
It is easy to show by induction that
$$F_m - 2 = F_0 F_1 \dots F_{m-1}.$$
This means that if $d$ divides both $F_n$ and $F_m$ (with $n < m$), then $d$ also divides $F_m - 2$.  Hence, $d$ divides 2.  But every Fermat number is odd, so $d$ is necessarily one.  This proves the lemma.
\end{proof}

We can now provide a solution to the exercise.

\begin{thm}[Goldbach, 1750]
There are infinitely many prime numbers.
\end{thm}

\begin{proof}
Choose a prime divisor $p_n$ of each Fermat number $F_n$.  By the lemma we know these primes are all distinct, showing there are infinitely many primes.
\end{proof}

\section{Demonstration of the environments}

We can format \emph{italic text}, \textbf{bold text}, and \texttt{code} blocks.

\begin{enumerate}
\item A numbered list item
\item Another numbered list item
\end{enumerate}

\begin{itemize}
\item A bulleted list item
\item Another bulleted list item
\end{itemize}

\begin{thm}
This is a theorem.  It contains an \texttt{align} block.

All math environments supported by MathJaX should work with LaTeX - a full list is available on the MathJaX homepage.

Maxwell's equations, differential form.
\begin{align*}
\nabla \cdot \mathbf{E} &= \frac {\rho} {\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}} {\partial t} \\
\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \\
\end{align*}
\end{thm}

\begin{thm}[Theorem name]
This is a named theorem.
\end{thm}

\begin{lem}
This is a lemma.
\end{lem}

\begin{prop}
This is a proposition
\end{prop}

\begin{proof}
This is a proof.
\end{proof}

\begin{lstlisting}
This is a code listing.
One line of code
Another line of code
\end{lstlisting}

\end{document}