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\begin{document}
\title{Summarized Results for 100,043\\ ``Safe" Prime\\Zero Omitted}
\date{\today}
\maketitle

\section*{General}
\noindent{\bf All: } 100,041 Graphs\\
{\bf Primitive Root: } 50,020 Graphs\\
{\bf Not Primitive Root: } 50,021 Graphs\\
\vspace{11 pt}\\
{\bf Largest Cycle: } 100,042 ($g = 20,812 \text{ and } 94,034$)\\
{\bf Longest Tail: } 1,448  ($g = 89,339$)\\
{\bf Shortest Maximum Cycle: } 1 ($g = 72,116 \text{ and } 91,980 \text{ and } 95,997$)\\

\section*{Number of Components}
\noindent{\bf Observed All: }
\begin{equation*}923,855/100,041 \approx 9.23\end{equation*}


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} \frac{1}{2}\log{n} =
\frac{1}{2}\log{100,043} \approx 5.76\end{equation*}\\

\noindent{\bf Observed Not PR: }
\begin{equation*}319,587/50,021 \approx 6.39\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}
\begin{equation*}\sum_{i=1}^{100,043} \frac{1}{i} \approx 12.09
\end{equation*}\\

\noindent{\bf Observed PR: }
\begin{equation*}604,268/50,020 \approx 12.08\end{equation*}




\section*{Number of Cyclic Nodes} \noindent{\bf Observed All:}
\begin{equation*}5,023,873,923/100,041\approx 50,218.15
\end{equation*}


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} \sqrt{\pi n / 2} = \sqrt{\pi100,043/2}
\approx 396.418\end{equation*}\\

\noindent{\bf Observed Not PR: }
\begin{equation*}19,773,083 / 50,021 \approx 395.296\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}100,042\end{equation*}

\noindent{\bf Observed PR:}
\begin{equation*}5,004,100,840 /50,020 = 100,042\end{equation*}


\section*{Number of Tail Nodes}
\noindent{\bf Observed All: }\begin{equation*}4,984,427,799
/100,041 \approx 49,823.85\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} n-\sqrt{\pi n / 2} = 100,042-\sqrt{\pi100,042/2}
\approx 99,645.58\end{equation*}\\

\noindent{\bf Observed Not PR: }
\begin{equation*}4,984,427,799 / 50,021 \approx 99,646.70\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical: }
\begin{equation*}0\end{equation*}
\noindent{\bf Observed PR:}\begin{equation*}0/50,020 =
0\end{equation*}\\



\section*{Number of Terminal Nodes}
\noindent{\bf Observed All:}\begin{equation*} 2,502,150,460
/100,041 \approx 25,011.25
\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*} e^{-1}n = e^{-1} * 100,042 \approx
36,803.395\end{equation*} \\

\noindent{\bf Observed Not PR:}\begin{equation*} 2,502,150,460
 / 50,021 = 50,021.99996
\end{equation*}
All values of $g$ except $100,042$ that were not PR had exactly
$50,022$.  $g=100,042$ had a total of $100,041$.  Since $100,042
\equiv -1 \mod{n}$ this exception makes sense.  In the other
cases, obviously, $g^0 \equiv g^{p-1} \equiv 1 \mod{n}$.

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*} 0 \end{equation*}

\noindent{\bf Observed PR:}\begin{equation*}
0 / 50,020 = 0\end{equation*} \\


\section*{Number of Image Nodes}
\noindent{\bf Observed All:}\begin{equation*}
100,043-25,012.25 \approx 75,030.75\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}(1-e^{-1})n = (1-e^{-1})*100,043 \approx
63,239.24\end{equation*} \\

\noindent{\bf Observed Not PR:}\begin{equation*} 100,043 -
50,022.99996 \approx 50,020.00\end{equation*}\\
The notes under Number of Terminal Nodes also apply here since
$\text{Terminal Nodes} + \text{Image Nodes} = n$.
\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}n-1=100,0437 - 1 =
100,042\end{equation*}\\

\noindent{\bf Observed PR:}\begin{equation*}
100,043 - 1 = 100,042\end{equation*} \\



\section*{Average Tail Length}
\noindent{\bf Observed All:}\begin{equation*}197.95
\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
\sqrt{\pi n/8} = \sqrt{\pi100,042/8}\approx 198.21
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*}197.96
\end{equation*}
Have to double check with new run that does not include 0.

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}0
\end{equation*}\\

\noindent{\bf Observed PR:}\begin{equation*}0
\end{equation*} \\





\section*{Average Cycle Length}
\noindent{\bf Observed All:}\begin{equation*}25,089.18
\end{equation*}\\

\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
\sqrt{\pi n/8} = \sqrt{\pi100,043/8}\approx 198.21
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*}198.315
\end{equation*}


\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}\frac{n+1}{2} =
\frac{100,043+1}{2} = 50,022
\end{equation*}\\

\noindent{\bf Observed PR:}\begin{equation*}49,980.6
\end{equation*} \\


\section*{Maximum Cycle Length}
\noindent{\bf Observed All:}\begin{equation*}31,321.1
\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
c_1\sqrt{n}\approx 0.78248\sqrt{100,043} \approx 247.495
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*}247.256
\end{equation*}

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*} 0.62432965n =
0.62432965*100043 \approx 62,495.81
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}62,395.5
\end{equation*} \\


\section*{Maximum Tail Length}
\noindent{\bf Observed All:}\begin{equation*}271.41
\end{equation*}\\


\subsection*{Maps}
\noindent{\bf Theoretical:}
\begin{equation*}
c_2\sqrt{n}\approx 1.73746\sqrt{100,042} \approx 549.55
\end{equation*}

\noindent{\bf Observed Not PR:}\begin{equation*}541.816
\end{equation*}
Have to double check with new run that does not include 0.

\subsection*{Permutations}
\noindent{\bf Theoretical:}\begin{equation*}0
\end{equation*}\\
\noindent{\bf Observed PR:}\begin{equation*}0
\end{equation*} \\



\end{document}
