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\author{Logan Axon}
\institute{Gonzaga University}
\date{\today}
\title{Logic basics}

\newtheorem{thm}{Theorem}
\theoremstyle{definition}
\newtheorem{dfn}{Definition}
\newtheorem{exercise}{Exercise}
\newtheorem{idea}{Idea}
\newtheorem{question}{Question}
\newtheorem{conjecture}{Conjecture}

\newcommand{\NN}{\mathbb{N}} % natural numbers
\newcommand{\ZZ}{\mathbb{Z}} % integers
\newcommand{\QQ}{\mathbb{Q}} % rational number
\newcommand{\RR}{\mathbb{R}} % real numbers

\newcommand{\pow}{\mathcal{P}} % power set


\begin{document}

%%%%%%
\begin{frame}{Perfect numbers}

\begin{dfn}
A natural number $n$ is \textbf{perfect} if it is equal to the sum of its proper divisors.
\end{dfn}

\pause

Examples:
\begin{align*}
6 &= 1 + 2+ 3 \\
28 &= 1 + 2 + 4 + 7 + 14 \\
496 &= 1 = 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
\end{align*}

Non-examples:
\begin{align*}
4 &\neq 1 + 2 = 3\\
9 &\neq 1 + 3 = 4\\
12 &\neq 1 + 2+ 3 + 4 + 6 = 16 
\end{align*}
\end{frame}

%%%%%%
\begin{frame}{Who cares?}

\begin{itemize}

\item Philo of Alexandria ($\sim20$ BCE--50 CE), a Jewish philosopher, claimed the world was created in 6 days that the moon orbits in 28 days because 6 and 28 are perfect.

\item Greek mathematicians like \href{http://www-groups.dcs.st-and.ac.uk/history/Biographies/Nicomachus.html}{Nichomachus} ($\sim100$ CE)  knew the perfect numbers on the last slide and the fourth: $8128$.

\item Egyptian mathematician  Ismail ibn Fall\={u}s (1194--1252 CE) knew the next three perfect numbers: $33,550,336$; $8,589,869,056$; and $137,438,691,328$.

\item Europeans were interested by the $15^{\text{th}}$ century when an unknown mathematician (re)discovered $33,550,336$.

\end{itemize}

\end{frame}

%%%%%%
\begin{frame}{And Euclid}

\begin{thm}[Euclid]
Let $n \in \mathbb{N}$.
If $\displaystyle 2^n - 1$ is prime, then $\displaystyle 2^{(n-1)} \left(2^n - 1\right)$ is perfect.
\end{thm}

\pause

\begin{dfn}
We define the following sets:
\begin{itemize}
\item $P$ is the set of perfect numbers;
\item $\displaystyle S = \left\{  2^{(n-1)} \left(2^n - 1\right): n \in \mathbb{N} \text{ and }  2^n - 1 \text{ is prime} \right\}$;
%\item $E$ is the set of even numbers.
\end{itemize}
\end{dfn}

\begin{question}
What relationship does Euclid's theorem establish between $S$ and $P$?
\end{question}

\end{frame}

\begin{frame}{Ibn al-Haytham and Euler too}

\begin{dfn}
We define the following sets:
\begin{itemize}
\item $P$ is the set of perfect numbers;
\item $\displaystyle S = \left\{  2^{(n-1)} \left(2^n - 1\right): n \in \mathbb{N} \text{ and }  2^n - 1 \text{ is prime} \right\}$;
\item $E$ is the set of even numbers.
\end{itemize}
\end{dfn}

\begin{conjecture}[Ibn al-Haytham, $\sim1000$ CE]
$( P \cap E) \subseteq S$.
\end{conjecture}

\pause

Proved by Euler in the $18^{\text{th}}$ century.

\begin{theorem}[Euclid-Euler]
$S = P \cap E$.
\end{theorem}

\end{frame}

\begin{frame}{Open questions}

\begin{question}
Are there infinitely many perfect numbers?
\end{question}

\pause

\begin{dfn}
A prime number of the form $\displaystyle 2^{n}-1$ is called a \textbf{Mersenne prime}.
\end{dfn}

\begin{question}
Are there infinitely many \href{https://www.mersenne.org/}{Mersenne primes}?
\end{question}

\pause

\begin{question} 
Are there any odd perfect numbers?
\end{question}

\end{frame}


\end{document}