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\begin{document}

\title{Say\i lar Kuram\i na Giri\c s}
\subtitle{Al\i\c st\i rmalar IV}
\author{David Pierce}
\date{24 Ekim 2017}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle
\thispagestyle{empty}

Tan\i ma g\"ore 
$n\in\N$,
$p$ asal,
$p\ndivides a$, ve $d\divides p-1$ olmak \"uzere
\begin{gather*}
\Z_n{}^{\times}=\{x\in\Z_n\colon\gcd(n,x)=1\},\\
\upvarphi(n)=\size{\Z_n{}^{\times}},\\
\ord pa=\min\{x\in\N\colon a^x\equiv1\}\pmod p,\\
\uppsi_p(d)=\size{\{x\in\Z_p{}^{\times}\colon\ord px=d\}}
\end{gather*}
olsun.  (Derste $\upvarphi(n)$, $\ord pa$, ve $\uppsi_p(d)$ tan\i mland\i\
ama $\Z_n{}^{\times}$ kullan\i lmad\i.)

\begin{exercise}
Her $p_i$ asal olmak \"uzere $p_1<p_2<\dots<p_n$ ise
$\sum_{i=1}^np_i{}^{-1}$
toplam\i n\i n bir tamsay\i\ olmad\i\u g\i n\i\ g\"osterin.
(Al\i\c st\i rmalar III ka\u g\i d\i ndaki bir teoremi kullanabilirsiniz.
\.Ilk olarak $n=2$ durumunu g\"osterebilirsiniz.)
\end{exercise}

\begin{exercise}
Hem $n$ hem $n+5$ asal olabilir mi?
\end{exercise}

\begin{exercise}
\begin{compactenum}
\item
E\u ger $n$ tek ise $\upvarphi(n)$ de\u gerinin \c cift oldu\u gunu g\"osterin.
(\emph{\.Ipucu}:  Baz\i\ tek $n$ i\c cin 
$\Z_n$ k\"umesi 
$\begin{array}{|*5{c|}}\hline1&2&3&\cdots&n\\\hline\end{array}$
% $\{1,2,3,\dots,n\}$ 
olarak yaz\i n
ve $\Z_n{}^{\times}$ altk\"umesinin elemanlar\i n\i\ se\c cin.)
\item
E\u ger $n$ \c cift ise $\upvarphi(n)$ tek midir?
\item
$\Z_n{}^{\times}$ k\"umesinin elemanlar\i n\i n toplam\i n\i n
$\bigl(n\cdot\upvarphi(n)\bigr)/2$
b\"ol\"um\"une e\c sit oldu\u gunu kan\i tlay\i n.
\end{compactenum}
\end{exercise}


\begin{exercise}
\begin{compactenum}
\item
$53$ mod\"ul\"une g\"ore a\c sa\u g\i daki gibi bir tabloyu \c cizip doldurun.
(Burada $(x+1)^2=x^2+2x+1$ e\c sitli\u gi faydal\i\ olabilir.)
\begin{comment}

\begin{gather*}
\begin{array}{|*{14}{r|}}\hline
x  &1&2&3& 4& 5&  6& 7&8&9&10&11&12&13\\\hline
x^2&1&4&9&16&25& 36&  &&&&&&\\\hline
x^2& & & &  &  &-17&-4&&&&&&\\\hline
\end{array}
\\
\begin{array}{|*{14}{r|}}\hline
x&14&15&16&17&18&19&20&21&22&23&24&25& 26\\\hline
x^2&&&&&&&&&&&&                      & 40\\\hline
x^2&&&&&&&&&&&&                      &-13\\\hline
\end{array}	
\end{gather*}

\end{comment}
\begin{equation*}
\begin{array}{|r|r|r|}\hline
x&x^2&x^2\\\hline
1&1&\\
2&4&\\
4&16&\\
5&25&\\
6&36&-17\\
7&  &-4\\
\multicolumn3{c}{\makebox[22mm]{\dotfill}}\\
%\dots&&\dots\\
26&40&-13\\\hline
\end{array}
\end{equation*}
\item
Tablonuzun sadece d\"ort sat\i r\i n\i\ kullanarak
$26$'n\i n $53$'\"un ilkel bir k\"ok\"u oldu\u gunu g\"osterin.
\end{compactenum}
\end{exercise}

\begin{exercise}
Hangi $p$ asal mod\"ullerine g\"ore
$a\equiv b$ kalanda\c sl\i\u g\i ndan $c^a\equiv c^b$ \c c\i kar?
\end{exercise}

\begin{exercise}
Hangi $p$ asal say\i lar\i\ i\c cin
$a$'n\i n $p$'nin ilkel bir k\"ok\"u oldu\u gu zamanda
$-a$ da $p$'nin ilkel bir k\"ok\"ud\"ur?
\end{exercise}

\begin{exercise}
$\uppsi_p(d)\leq\upvarphi(d)$ kan\i tlay\i n.
(Derste $p=13$ durumuna bakt\i k.)
\end{exercise}


\end{document}
