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\begin{document}

\title{Say\i lar Kuram\i na Giri\c s}
\subtitle{\"Ozet}
\author{David Pierce}
\date{26 Aral\i k 2017}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle
\thispagestyle{empty}

\section{Toplama ve s\i ralama}

S\i n\i rs\i z bir do\u gruda, bir nokta $0$ olarak se\c cilirse,
o zaman
\"Oklid'in 3.\ \"onermesi ile iki noktan\i n \textbf{toplam\i}
ve bir noktan\i n \textbf{negatifi} tan\i mlanabilir.
Sonu\c c olarak do\u grunun herhangi $a$, $b$, ve $c$ noktalar\i\ i\c cin
\begin{gather*}
  a+(b+c)=(a+b)+c,\\
a+b=b+a,\\
a+0=a,\\
a+(-a)=0.
\end{gather*}
Tan\i ma g\"ore
\begin{equation*}
  a-b=a+(-b).
\end{equation*}
E\u ger bir $b$ noktas\i\ bir $a$ noktas\i n\i n sa\u g\i ndaysa,
o zaman $a$, $b$'den \textbf{k\"u\c c\"uk}
ve $b$, $a$'dan \textbf{b\"uy\"uk} olarak say\i l\i r,
ve
\begin{align*}
  a&<b,&b&>a
\end{align*}
yaz\i l\i r.
Tan\i ma g\"ore
\begin{gather*}
a\leq a,\\
  a<b\implies a\leq b,\\
a\leq b\iff b\geq a.
\end{gather*}
O halde
\begin{equation}\label{eqn:ord}
\left.\qquad
\begin{gathered}
a<b\implies a\neq b,\\
  a\leq b\And b\leq a\implies a=b,\\
a\leq b\And b\leq c\implies a\leq c.
\end{gathered}
\qquad\right\}
\end{equation}
Ayr\i ca
\begin{equation*}
  a<b\implies a+c<b+c,
\end{equation*}
dolay\i s\i yla
\begin{equation*}
  a<b\iff b-a>0.
\end{equation*}

\section{Sayma say\i lar\i}

$0$'\i n sa\u g\i nda olan bir nokta $1$ olarak se\c cilsin.
%\c Simdi do\u grunun noktalar\i\ \textbf{ger\c cel say\i lar} say\i l\i r.
\textbf{Sayma say\i lar\i n\i n} \"ozyineli tan\i m\i na g\"ore
\begin{compactenum}[(i)]
  \item
$1$ bir sayma say\i s\i d\i r, ve
\item
e\u ger $n$ bir sayma say\i s\i\ ise,
o zaman $n+1$ de bir sayma say\i s\i d\i r.
\end{compactenum}
Burada $n+1$, $n$'nin \textbf{ard\i l\i d\i r.}
Sayma say\i lar\i\ k\"umesi
$\N$ olarak yaz\i l\i r.
O halde
\begin{equation*}
  \N=\{1,2,3,\dots\}.
\end{equation*}
Tan\i mdan $\N$'nin herhangi $A$ altk\"umesi i\c cin,
e\u ger
\begin{compactenum}[(i)]
  \item
$1\in A$ ise ve
\item
$A$ her eleman\i n\i n ard\i l\i n\i\ da i\c cerirse,
\end{compactenum}
o zaman $A=\N$.
Bu sonu\c c, \textbf{T\"umevar\i m \.Ilkesidir.}
\begin{compactitem}
  \item
T\"umevar\i m, 
\item
$\N$'n\i n s\i ralanmas\i n\i n \eqref{eqn:ord}
\"ozellikleri ve 
\item
$n<n+1$ kural\i ndan
\end{compactitem}
$\N$'nin b\"ut\"un \"ozellikleri elde edilebilir.

\"Ozellikle \textbf{\"Ozyineleme Teoremi} kan\i tlanabilir
(ama kan\i tlamad\i k).
Bu teoreme g\"ore, 
e\u ger $B$ bir k\"ume ise, $c\in B$ ise, ve $f$,
$B$'nin tek konumlu bir i\c slemi 
(yani $f\colon B\to B$) ise,
o zaman $\N$'den $B$'ye giden bir ve tek bir $h$ g\"ondermesi i\c cin
\begin{compactenum}[(i)]
  \item
$h(1)=c$,
\item
her $n$ sayma say\i s\i\ i\c cin $h(n+1)=f(h(n))$.
\end{compactenum}
\"Orne\u gin $\N$'de toplama
\begin{align*}
  m+1&=(\text{$m$'nin ard\i l\i}),&m+(n+1)=(m+n)+1
\end{align*}
kurallar\i\ ile tan\i mlanabilir.
T\"umevar\i m ile,
toplaman\i n g\"ord\"u\u g\"um\"uz \"ozellikleri kan\i tlanabilir.

Al\i\c st\i rmalar I'deki gibi do\u grumuzun noktalar\i n\i n,
sayma say\i lar\i\ ile 
\textbf{katlar\i} (yani \c co\u galt\i lmalar\i)
ve \textbf{kuvvetleri} tan\i mlan\i r,
ve bunlar\i n \"ozellikleri t\"umevar\i m ile kan\i tlan\i r.

E\u ger do\u grunun $0<a<b$ e\c sitsizli\u gini sa\u glayan 
herhangi $a$ ve $b$ noktalar\i\ i\c cin,
bir $n$ sayma say\i s\i\ i\c cin, $b<a\cdot n$ ise,
o zaman do\u grunun \textbf{Ar\c simet \"Ozelli\u gi} vard\i r,
ve bu durumda do\u grunun noktalar\i\
\textbf{ger\c cel say\i lar} olarak say\i labilir.
Asl\i nda Ar\c simet \"Ozelli\u gi'ni kullanmayaca\u g\i z.

\c Co\u galtma ile
$\N$'de iki konumlu \textbf{\c carpma} i\c slemini elde ederiz,
ve $\N$'de
\begin{equation*}
  k\cdot m=m\cdot k
\end{equation*}
\"ozelli\u gi, t\"umevar\i m ile kan\i tlan\i r.

\"Ozyineli tan\i ma g\"ore
\begin{align*}
\sum_{k=1}^1a_k&=a_1,&\sum_{k=1}^{n+1}&=\sum_{k=1}^na_k+a_{n+1}.
\end{align*}
Benzer \c sekilde
\begin{align*}
\prod_{k=1}^1a_k&=a_1,&\prod_{k=1}^{n+1}&=\prod_{k=1}^na_k\cdot a_{n+1}.
\end{align*}

Sayma say\i lar\i\ ve $0$, \textbf{do\u gal say\i lard\i r.}
Tan\i ma g\"ore $\omega$ (omega), do\u gal say\i lar\i n k\"umesidir:
\begin{equation*}
  \upomega=\{0\}\cup\N=\{0,1,2,\dots\}.
\end{equation*}
Bazen
\begin{align*}
\sum_{k=1}^0a_k&=0,&\prod_{k=1}^0a_k&=1
\end{align*}
kurallar\i na ihtiyac\i m\i z vard\i r.
\"Orne\u gin tan\i ma g\"ore $n\in\upomega$ ise
\begin{equation*}
  n!=\prod_{k=1}^nk.
\end{equation*}
Daha fazla \"ornekler i\c cin,
Al\i\c st\i rmalar I ve V'e bak\i n.

\section{B\"olme}

$\N$'de $a\cdot b=c$ ise $a$ ve $b$,
$c$'nin \textbf{\c carpan\i} veya \textbf{b\"olenidir,}
ve her biri $c$'yi \textbf{b\"oler.}
$\N$'de $p>1$ ise ve $p$'nin $1$'den ve kendisinden farkl\i\ olan
hi\c c \c carpan\i\ yoksa, $p$ \textbf{asald\i r.}

\textbf{\.Iyis\i ralama Teoremine} g\"ore,
herhangi verilen sayma say\i lar\i ndan biri,
onlar\i n en k\"u\c c\"u\u g\"ud\"ur.
Sonu\c c olarak $1$'den b\"uy\"uk olan her sayma say\i s\i n\i n
asal bir \c carpan\i\ vard\i r.
Asl\i nda verilen say\i n\i n $1$'den b\"uy\"uk olan \c carpanlar\i n\i n
en k\"u\c c\"u\u g\"u asald\i r.
Al\i\c st\i rma I'e bak\i n.

\textbf{G\"u\c cl\"u T\"umevar\i m Teoremine} g\"ore,
$\N$'nin herhangi $A$ altk\"umesi i\c cin, 
e\u ger her $n$ sayma say\i s\i\ i\c cin
\begin{equation*}
  \{x\in\N\colon x<n\}\included A\implies n\in A
\end{equation*}
ise, o zaman $A=\N$.
\"Orne\u gin her sayma say\i s\i n\i n
\textbf{asal \c carpanlara ayr\i l\i\c s\i} vard\i r.
Bu ayr\i l\i\c s, $p_1\leq\cdots\leq p_m$
ve her biri asal olmak \"uzere
\begin{equation*}
  \prod_{k=1}^mp_k
\end{equation*}
\c seklinde yaz\i labilir.
Zira bir $n$ i\c cin 
$n$'den k\"u\c c\"uk olan her sayma say\i s\i\ i\c cin 
iddia do\u gru olsun.
E\u ger $n$ asal veya $1$'e e\c sit ise,
o zaman a\c sik\^ar bir \c sekilde
$n$'nin asal \c carpanlara ayr\i l\i\c s\i\ vard\i r.
E\u ger $n>1$ ise ama $n$ asal de\u gilse, 
$1$'e e\c sit olmayan baz\i\ $a$ ve $b$ i\c cin
\begin{equation*}
  n=a\cdot b.
\end{equation*}
Varsay\i ma g\"ore $a$ ve $b$'den her birinin
asal \c carpanlara ayr\i l\i\c s\i\ vard\i r,
ve bunlardan $n$'nin asal \c carpanlara ayr\i l\i\c s\i\ elde edilir.
(Ayn\i\ sonu\c c, 
Al\i\c st\i rma I'deki gibi iyis\i ralama ile kan\i tlanabilir.)

\textbf{B\"olme Teoremine} g\"ore $\N$'de
herhangi $a$ ve $b$ i\c cin ya
\begin{equation*}
  a=bx
\end{equation*}
denklemi ya da
\begin{equation*}
  a=bx+y\And y<b
\end{equation*}
sistemi \c c\"oz\"ulebilir.
Bu teoremden
(ve $\N$'nin iyis\i ral\i\ oldu\u gundan)
\textbf{\"Oklid Algoritmas\i} ile
iki say\i n\i n
\textbf{en b\"uy\"uk ortak b\"oleni} bulunur.
Al\i\c st\i rma II'ye bak\i n.
Ba\c ska bir teoreme g\"ore
$a$ ve $b$'nin \textbf{en k\"u\c c\"uk ortak kat\i} vard\i r ve
\begin{equation*}
  a\cdot b=\gcd(a,b)\cdot\lcm(a,b).
\end{equation*}

\section{Tamsay\i lar}

Sayma say\i lar\i, negatifleri ve $0$,
\textbf{tamsay\i lard\i r.}
Bunlar\i n k\"umesi $\Z$ olarak yaz\i l\i r.
Buradaki toplama ve \c carpman\i n temel kurallar\i,
Al\i\c st\i rma II'nin Al\i\c st\i rma 1'indedir.
\textbf{B\'ezout Lemmas\i'na} g\"ore
\begin{equation*}
  ax+by=\ebob(a,b)
\end{equation*}
denklemi $\Z$'de \c c\"oz\"ulebilir.
Bir \c c\"oz\"um,
\"Oklid Algoritmas\i'n\i n ad\i mlar\i ndan elde edilebilir.

B\'ezout Lemmas\i'ndan
\textbf{\"Oklid Lemmas\i}
ve Al\i\c st\i rmalar III'teki genelle\c stirilmesi elde edilir.
\"Oklid Lemmas\i\ sayesinde
her sayma say\i s\i n\i n asal \c carpanlara ayr\i l\i\c s\i\ tektir;
bu sonu\c c, \textbf{Temel Aritmetik Teoremidir.}

$\Z$'de tan\i ma g\"ore
\begin{equation*}
  a\divides b\iff\text{ bir $x$ i\c cin }ax=b.
\end{equation*}
O zaman
\textbf{Fermat Teoremine} g\"ore her $p$ asal\i\ i\c cin
\begin{equation}\label{thm:Fermat1}
  p\divides a^p-a,
\end{equation}
ve ayr\i ca
\begin{equation}\label{thm:Fermat2}
  p\ndivides a\implies p\divides a^{p-1}-1.
\end{equation}
Ayn\i\ teorem, \textbf{kalanda\c sl\i klar} ile ifade edilebilir:
\begin{gather*}
  a^p\equiv a\pmod p,\\
  p\ndivides a\implies a^{p-1}\equiv1\pmod p.
\end{gather*}
\c Simdi $p\ndivides a$ olsun.
O zaman $\N$'de
\begin{equation*}
  a^x\equiv1\pmod p
\end{equation*}
kalanda\c sl\i\u g\i n\i n \c c\"oz\"um\"u vard\i r.
En k\"u\c c\"uk \c c\"oz\"um, $a$'n\i n $p$'ye g\"ore
\textbf{mertebesidir} (\.Ingilizce \emph{order}).
Bu mertebe $m$ ise, o zaman $m\divides p-1$,
ve ayr\i ca, $p-1=mn$ olmak \"uzere
baz\i\ $b_1$, \dots, $b_{n-1}$ say\i lar\i\ i\c cin
her tamsay\i, a\c sa\u g\i daki matrisin bir ve tek bir girdisine
$p$'ye g\"ore denktir:
\begin{equation*}
  \begin{bmatrix}
1     &      a&      a^2&\cdots&   a^{m-1}\\
b_1   &   b_1a&b_1   a^2&\cdots&b_1a^{m-1}\\
\hdotsfor5\\
b_{n-1}&b_{n-1}a&b_{n-1}a^2&\cdots&b_{n-1}a^{m-1}
  \end{bmatrix}
\end{equation*}
(Bu sonuca \textbf{Lagrange Teoremi} diyebiliriz
ama derste demedik.)
\"Orne\u gin $p=13$ durumunda
\begin{equation*}
  \begin{bmatrix}
    1&3&9\\
2&6&18\\
4&12&36\\
7&21&63
  \end{bmatrix}
\end{equation*}
matrisi \c c\i kar.
Bunu daha iyi anlamak i\c cin,
girdilerin yerine
ya $\{1,\dots,12\}$
ya da $\{-6,\dots,-1\}\cup\{1,\dots,6\}$ k\"umesinde olan,
$13$'e g\"ore denk olan say\i lar\i\ koyabiliriz:
\begin{align*}
  &\begin{bmatrix}
    1&3&9\\
2&6&5\\
4&12&10\\
7&8&11
  \end{bmatrix}&
  &\begin{bmatrix}
    1&3&-4\\
2&6&5\\
4&-1&-3\\
-6&-5&-2
  \end{bmatrix}
\end{align*}

Verilen bir mod\"ule g\"ore denklik,
bir \textbf{denklik ba\u g\i nt\i s\i d\i r,}
\c c\"unk\"u yans\i mal\i, simetrik, ve ge\c ci\c slidir.
\"Oklid i\c cin s\i n\i rl\i\ do\u grular\i n e\c sitli\u gi
bir denklik ba\u g\i nt\i s\i d\i r.
Bizim i\c cin tan\i ma g\"ore
\begin{equation*}
  (a,b)\sim(c,d)\iff ad=bc
\end{equation*}
ise, o zaman $\N\times\N$'de $\sim$ ba\u g\i nt\i s\i\
bir denklik ba\u g\i nt\i s\i d\i r.

Bir $A$ k\"umesinde $E$ bir denklik ba\u g\i nt\i s\i\ ise,
o zaman $A$'n\i n her $b$ eleman\i n\i n
($E$'ye g\"ore) \textbf{denklik s\i n\i f\i}
\begin{equation*}
  \{x\in A\colon b\mathrel Ex\}
\end{equation*}
k\"umesidir.  Bu k\"ume i\c cin $[b]$ yaz\i ls\i n.
O zaman
\begin{equation*}
  [b]=[c]\iff b\mathrel Ec.
\end{equation*}
\"Orne\u gin $\sim$ ba\u g\i nt\i s\i\ yukar\i daki gibi ise,
$[(a,b)]$ s\i n\i f\i\ $a/b$ kesirli say\i s\i\ olarak anla\c s\i labilir.

Genelde $A$'n\i n elemanlar\i n\i n denklik s\i n\i flar\i\
bir $A/E$ k\"umesini olu\c sturur.
E\u ger $A=\Z$ ise ve $E$, bir $n$ mod\"ul\"une g\"ore denklik ise,
o zaman $A/E$,
\begin{equation*}
  \Z_n
\end{equation*}
olarak yaz\i labilir.
Bu k\"ume $\{1,\dots,n\}$, $\{0,\dots,n-1\}$ veya
($n=2m+1$ durumunda) $\{-m,\dots,m\}$ olarak anla\c s\i labilir.
O zaman tan\i ma g\"ore
\begin{equation*}
  \units{\Z_n}=\{x\in\Z_n\colon\ebob(n,x)=1\}.
\end{equation*}
B\'ezout Lemmas\i\ ile bu k\"umenin her eleman\i n\i n tersi bulunabilir.
Bu nedenle Fermat Teoreminin \eqref{thm:Fermat2} par\c cas\i,
\eqref{thm:Fermat1} par\c cas\i ndan \c c\i kar.

Tan\i ma g\"ore $\upphi(n)$, 
$\units{\Z_n}$ k\"umesinin elemanlar\i n\i n say\i s\i d\i r.
O zaman \textbf{Gauss Teoremine} g\"ore
\begin{equation*}
  \sum_{d\divides n}\upphi(d)=n.
\end{equation*}
\textbf{Euler Teoremine} g\"ore 
$\gcd(n,a)=1$ ise $a^{\upphi(n)}\equiv1\pmod n$,
ama bunu g\"ostermedik;
\c co\u gunlukla asal mod\"uller ile \c cal\i\c smay\i\ tercih ederiz.

E\u ger tekrar $p$ asal ise,
o zaman $\units{\Z_p}$ k\"umesinde 
$a^n$ kuvvetlerini hesaplamak i\c cin
bir y\"ontem vard\i r:
\begin{compactenum}
\item
\"Oyle $m$'yi bulun ki $m\equiv n\pmod{p-1}$ ve 
$\frac12(p-1)<m\leq\frac12(p-1)$ olsun.
O zaman $a^n\equiv a^m\pmod p$.
\item
E\u ger $m<0$ ise $m$'nin yerine $-m$'yi,
$a$'nin yerine $a^{-1}$'i kullan\i n.
  \item
$2$'nin farkl\i\ kuvvetlerinin bir toplam\i\ olarak $m$'yi yaz\i n.
\item
Buradaki $2$'nin en y\"uksek kuvveti $2^{\ell}$ ise
ad\i m ad\i m $a^2$, $a^{2^2}$, \dots, $a^{2^{\ell}}$ 
kuvvetlerini $p$'ye g\"ore hesaplay\i n.
\item
Gereken \c carp\i mlar\i\ \c carparak $a^m$'yi elde edin.
\end{compactenum}

\c Simdi $p$'ye g\"ore bir $a$'n\i n mertebesi
\begin{equation*}
  \ord pa
\end{equation*}
olarak yaz\i ls\i n.  Bu mertebe $m$ ise
\begin{equation*}
  \ord p{a^k}
=\frac{\lcm(m,k)}k
=\frac m{\gcd(m,k)}.
\end{equation*}
E\u ger $\ord pa=p-1$ ise,
o zaman $a$'ya $p$'nin \textbf{ilkel bir k\"ok\"u} denir.
Kan\i tlad\i\u g\i m\i z bir teoreme g\"ore 
her asal say\i n\i n ilkel bir k\"ok\"u vard\i r.
Asl\i nda $d\divides p-1$ ise
$\upphi(d)$,
$\units{\Z_p}$ k\"umesinin
mertebesi $n$ olan elemanlar\i n\i n say\i s\i d\i r.

E\u ger $a$, $p$'nin ilkel bir k\"ok\"u ise,
o zaman $p$'ye g\"ore
\begin{multline*}
(p-1)!
\equiv\prod_{k=1}^{p-1}k
\equiv\prod_{k\in\units{\Z_p}}k
\equiv\prod_{j\in\Z_{p-1}}a^j
\equiv\prod_{j=1}^{p-1}a^j\\
\equiv a^{\sum_{j=1}^{p-1}j}
\equiv a^{p\cdot(p-1)/2}
\equiv(a^p)^{(p-1)/2}
\equiv a^{(p-1)/2}
\equiv-1.
\end{multline*}
Bu sonu\c c, \textbf{Wilson Teoremidir.}

\c Simdi $\ebob(k,m)=1$ olsun.
E\u ger $mx+ky=1$ ise,
o zaman $amx+bky$,
\begin{align*}
  t&\equiv a\pmod k,&t&\equiv b\pmod m.
\end{align*}
sistemini \c c\"ozer.
E\u ger tersine $d$ ve $e$ bu sistemi \c c\"ozerse,
o zaman
(Al\i\c st\i rma V'ten)
$d\equiv e\pmod{km}$.
Bu sonu\c c, \textbf{\c Cin Kalan Teoremidir.}
E\u ger ayr\i ca $\ebob(km,n)=1$ ve
\begin{align*}
  mnx&\equiv1\pmod k,\\
  kny&\equiv1\pmod m,\\
  kmz&\equiv1\pmod n
\end{align*}
ise, o zaman
\begin{align*}
  t&\equiv a\pmod k,&
  t&\equiv b\pmod m,&
  t&\equiv c\pmod n
\end{align*}
sisteminin \c c\"oz\"umleri,
\begin{equation*}
  t\equiv amnx+bkny+ckmz\pmod{kmn}
\end{equation*}
kalanda\c sl\i\u g\i n\i n  \c c\"oz\"umleridir.

\end{document}
