Skip to main content
Contents Index Dark Mode Prev Up Next \(\newcommand{\im}{{\mathrm{im}}}
\newcommand{\given}{{\, :\,}}
\newcommand{\hanoi}{\mathrm{hanoi}}
\newcommand{\lcm}{{\mathrm{lcm}}}
\newcommand{\ID}{{\mathrm{ID}}}
\newcommand{\chr}{{\mathtt{chr}}}
\newcommand{\ord}{{\mathtt{ord}}}
\newcommand{\set}[1]{\left\lbrace #1 \right\rbrace}
\newcommand{\seq}[2]{( #1 )_{#2}}
\newcommand{\gr}[1]{\left( #1 \right)}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\size}[1]{\# #1}
\newcommand{\id}[1]{\mathrm{id}_{#1}}
\newcommand{\ip}[1]{\left\langle #1 \right\rangle}
\newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}
\newcommand{\ceiling}[1]{\left\lceil #1\right\rceil}
\newcommand{\indemph}[1]{{\textbf{\emph{#1}}\index{#1}}}
\renewcommand{\implies}{\to}
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\lxor}{\oplus}
\newcommand{\cP}{\wp}
\newcommand{\pset}{\cP}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\barr}[1]{\overline{#1}}
\newcommand{\sage}{\textsc{Sage}}
\newcommand{\magma}{\textsc{Magma}}
\newcommand{\bob}{\text{Bob}}
\newcommand{\true}{\text{T}}
\newcommand{\false}{\text{F}}
\newcommand{\ahat}{\hat{a}}
\newcommand{\ah}{\ahat}
\newcommand{\bhat}{\hat{b}}
\newcommand{\abar}{\bar{a}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\todo}[1]{{\color{red} TODO: #1}}
\renewcommand{\div}{\operatorname{div}}
\newcommand{\gexp}[3]{#1^{#2 #3}}
\newcommand{\gexpp}[3]{\displaystyle\left(#1\right)^{#2 #3}}
\newcommand{\glog}[3]{\log_{#1}^{#3}#2}
\def\dbend{{!!!}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Chapter 5 Number Theory and Applications
Mathematics is the queen of the sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
―Carl Friedrich Gauss (1777--1855)
At its core,
number theory is the study of integers. In this chapter, we examine introduce modular arithmetic to explore divisibility. We discuss modern applications of number theory including ASCII encoding and encryption.
