\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Fermat&#39;s little theorem}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{refinements}{Refinements}\dotfill \pageref*{refinements} \linebreak
\noindent\hyperlink{pseudoprimes_and_carmichael_numbers}{Pseudoprimes and Carmichael numbers}\dotfill \pageref*{pseudoprimes_and_carmichael_numbers} \linebreak
\noindent\hyperlink{aks_primality_test}{AKS primality test}\dotfill \pageref*{aks_primality_test} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

For $x$ an [[integer]] and $p$ a [[prime number]], if $x \nequiv 0 \pmod{p}$, then $x^{p-1} \equiv 1 \pmod{p}$.

This useful result is in a sense trivial, since the [[ring]] $\mathbb{Z}/(p)$ is a finite [[field]], with [[group of units]] $G$ of [[order of a group|order]] $p-1$; it is just a matter of recalling $x^{ord(G)} = 1$ for all $x$ in a [[finite group]] $G$. For the same reason, $a^{q-1} = 1$ for any nonzero $a$ in a finite field with $q$ elements.

\hypertarget{refinements}{}\subsection*{{Refinements}}\label{refinements}

A stronger result is that the group of units of a finite field with $q$ elements is \emph{[[cyclic group|cyclic]]} of order $q-1$, or indeed that every finite subgroup of the group of units of any field is cyclic. A proof may be found \href{/nlab/show/root#rootsunity}{here}.

If $A$ is a commutative $\mathbb{F}_p$-[[associative unital algebra|algebra]], then the map $a \mapsto a^p$ is an algebra endomorphism $\sigma: A \to A$ (the preservation of addition follows easily from the [[binomial theorem]] and the fact that $\binom{p}{k} \equiv 0 \pmod{p}$ when $0 \lt k \lt p$; apparently this is also known as the [[freshman's dream]]). It follows that $\sigma: \mathbb{F}_q \to \mathbb{F}_q$ is a field \emph{automorphism} on a field with $q = p^k$ elements, since after all

\begin{displaymath}
\sigma^k(a) = a^{p^k} = a^q = a^{q-1}a = a
\end{displaymath}
for any $a \in \mathbb{F}_q$. See also [[Frobenius automorphism]].

\hypertarget{pseudoprimes_and_carmichael_numbers}{}\subsection*{{Pseudoprimes and Carmichael numbers}}\label{pseudoprimes_and_carmichael_numbers}

Fermat's little theorem says that in order for a positive integer $p$ to be prime, it is necessary that $a^p \equiv a \pmod{p}$ for any integer $a$ (one may as well assume $0 \leq a \lt p$). This gives a way of showing that an integer $n$ is \emph{not} prime (by finding an $a$ less than $n$ such that $a^{n-1} \nequiv 1 \pmod{n}$) that, especially for large $n$, is more efficient than actually factoring $n$.

One type of (probabilistic) primality test for $p$ is to take a base, for example $a = 2$, and check whether $a^{p-1} = 1 \pmod{p}$. Chances are good that $p$ is in fact prime if this is satisfied, although there certainly exist composite numbers which pass this test (called pseudoprimes base $a$). The smallest pseudoprime base $2$ is $341 = 11 \cdot 31$. One effective primality test for ``small'' $p$ (e.g., less than $10^{15}$) is to use such a primality test coupled with a table of pseudoprimes.

There are numbers such as $n = 561 = 3 \cdot 11 \cdot 17$ which are pseudoprimes in any base; these are called \emph{Carmichael numbers}. A positive integer $n$ is Carmichael iff it is square-free and for each prime divisor $p$ of $n$, we have that $p-1$ is a divisor of $n-1$. They are comparatively rare, but it is known there are infinitely many (for sufficiently large $n$, there are at least $n^{2/7}$ Carmichael numbers between $1$ and $n$).

\hypertarget{aks_primality_test}{}\subsection*{{AKS primality test}}\label{aks_primality_test}

A generalization of Fermat's little theorem can be used to give a deterministic test for primality that can be carried out on any integer $n$ in ``polynomial time'' (bounded by a polynomial applied to the number of digits of $n$); it was first published only as recently as 2002.

\begin{lemma}
\label{}\hypertarget{}{}
An integer $n \geq 2$ is prime if and only if

\begin{displaymath}
(x-a)^n \equiv x^n - a \pmod{n}
\end{displaymath}
for every or even any $a$ coprime to $n$ (so that, for example, the case $a=1$ would be a sufficient criterion for primality).

\end{lemma}
\begin{proof}
By the \href{nlab/show/freshman's+dream#primality}{argument at freshman's dream}, primality of $n$ is equivalent to $(x-a)^n \equiv x^n - a^n \pmod{n}$ (for every or any $a$), and then primality of $n$ implies the further reduction $a^n \equiv a \pmod{n}$ by Fermat's little theorem.

\end{proof}
Given an integer $r$, and an integer $a$ relatively prime to $r$, let $ord_r(a)$ denote the order of $a \pmod{r}$ as a unit in $\mathbb{Z}/(r)$. We let $log(n)$ denote the base $2$ [[logarithm]] of $n$, and $\phi$ denotes the [[Euler totient function]] (so $\phi(r)$ is the cardinality of the [[group of units]] of $\mathbb{Z}/(r)$).

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Agrawal, Kayal, Saxena)} For given $n$, let $r$ be the least positive integer such that $ord_r(n) \gt (log(n))^2$. Then $n$ is prime if and only if either

\begin{itemize}%
\item $n \leq r$ and whenever $1 \leq a \leq r$, either $1 = \gcd(a, n)$ or $n = \gcd(a, n)$, or


\item $r \lt n$ and whenever $a$ is an integer such that $1 \leq a \leq \sqrt{\phi(r)} \cdot log(n)$, we have

\begin{displaymath}
(x - a)^n \equiv x^n - a \pmod{x^r - 1, n}.
\end{displaymath}


\end{itemize}
\end{theorem}
From this result, one can extract an algorithm that decides primality of $n$ in time bounded by a polynomial in $log(n)$, invoking help from the following result.

\begin{lemma}
\label{}\hypertarget{}{}
For each $n$ there exists $r \leq \max \{3, (log(n))^5\}$ such that $ord_r(n) \gt (log(n))^2$.

\end{lemma}
These results form the basis for the first algorithm for deciding primality that is general (applies to any integer $n$), runs in polynomial time, is deterministic (other known efficient tests were randomized and only guaranteed high probability of primality), and unconditional (does not depend on conjectured number-theoretic results such as forms of the [[Riemann hypothesis]]).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Fermat quotient]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after \emph{[[Pierre de Fermat]]}.

\begin{itemize}%
\item Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, \emph{PRIMES is in P}, Annals of Mathematics 160 (2) (2004), 781--793. doi:10.4007/annals.2004.160.781. JSTOR 3597229. (\href{http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf}{pdf})

\end{itemize}


\end{document}