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\begin{document}

\title{Title }
\author{ Doron Zeilberger}

\maketitle

\begin{abstract}
We prove RH and $3x+1$
\end{abstract}
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\section*{Supporting Maple packages and output}

All the results in this article were obtained by the use of the Maple packages

$\bullet$ {\tt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/Armin.txt} \quad ,

$\bullet$ {\tt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/core.txt} \quad ,

whose output files, along with links to diagrams, are available from the {\it front} of this article

{\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/armin.html}.

 \section{Section 1}

Blah

\section{Section 2}


\subsection{}

Blah

\begin{figure}
\centering
\begin{subfigure}[t]{0.7\textwidth}
\includegraphics[width=\textwidth]{1a.jpg}
\caption{The lattice $P_{13,15}$.}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\includegraphics[width=\textwidth]{1b.jpg}
\caption{An order ideal of $P_{13,15}$ with smallest unoccupied odd label 5 and no consecutive labels must be a subset of the above lattice.}
\end{subfigure}
\caption{}
\end{figure}


\begin{figure}
\centering
\begin{subfigure}[t]{0.6\textwidth}
\includegraphics[width=\textwidth]{2a.jpg}
\caption{The lattice $EO(3,6)$.}
\end{subfigure}
\begin{subfigure}[b]{0.6\textwidth}
\includegraphics[width=\textwidth]{2b.jpg}
\caption{An order ideal of $EO(3,6)$ with smallest unoccupied odd label 5 and no consecutive labels must be a subset of the above lattice, which is isomorphic to $OE(1,3)$ union the triangular component containing the  labels $1,3$.}
\end{subfigure}
\caption{}
\end{figure}


\subsection{Encore:}

\section*{Acknowledgment}

We wish to thank Tewodros Amdeberhan for introducing us to this interesting subject. 


\section*{References}

[Am] Tewodros Amdeberhan, {\it Theorems, problems and conjectures},\\ {\tt https://arxiv.org/abs/1207.4045} \quad .



\bigskip
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Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail dot com}   \quad .


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