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\title{Number-theory exercises, XI}
\author{David Pierce}
\date{\today}

\address{Mathematics Dept\\
Middle East Tech.\ Univ.\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}


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\begin{document}
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  \begin{xca}
    Compute the Legendre symbols $(91/167)$ and $(111/941)$.
  \end{xca}

  \begin{xca}
Find $(5/p)$ in terms of the class of $p$ \emph{modulo}
$5$.  
  \end{xca}

  \begin{xca}
    Find $(7/p)$ in terms of the class of $p$
    \emph{modulo} $28$.
  \end{xca}

  \begin{xca}
The $n$th \defn{Fermat number}, or $F_n$, is $2^{2^n}+1$.  A
\defn{Fermat prime}{} is a Fermat number that is prime.
\begin{enumerate}
  \item
Show that every prime number of the form $2^m+1$ is a Fermat prime.
\item
Show $4^k\equiv4\pmod{12}$ for all positive $k$.
\item
If $p$ is a Fermat prime, show $(3/p)=-1$.
\item
Show that $3$ is a primitive root of every Fermat prime.
\item
Find a prime $p$ less than $100$ such that $(3/p)=-1$, but $3$ is not
a primitive root of $p$.
\end{enumerate}
  \end{xca}

  \begin{xca}
    Solve the congruence $x^2\equiv11\pmod{35}$.
  \end{xca}

  \begin{xca}
We have so far defined the Legendre symbol $(a/p)$ only when
    $p\ndivides a$; but if $p\divides a$, then we can define $(a/p)=0$.
    We can now define $(a/n)$ for
    arbitrary $a$ and $n$: the result is the \defn{Jacobi symbol}, and
    the definition is
    \begin{equation*}
      \ls an=\prod_p\ls ap^{k(p)},\quad\text{ where }\quad
      n=\prod_pp^{k(p)}. 
    \end{equation*}
    \begin{enumerate}
      \item
Prove that the function $x\mapsto(x/n)$ on $\Z$ is \defn{completely
  multiplicative}{} in the sense that $(ab/n)=(a/n)\cdot(b/n)$ for all
  $a$ and $b$ (not necessarily co-prime).
\item
If $\gcd(a,n)=1$, and the congruence $x^2\equiv a\pmod n$ is soluble,
show $(a/n)=1$.
\item
Find an example where $(a/n)=1$, and $\gcd(a,n)=1$, but $x^2\equiv
a\pmod n$ is insoluble.
\item
If $m$ and $n$ are co-prime, show
\begin{equation*}
  \ls mn\cdot\ls nm=(-1)^k,\quad\text{ where }\quad
  k=\frac{m-1}2\cdot\frac{n-1}2. 
\end{equation*}
    \end{enumerate}
  \end{xca}


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