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

\address{Mathematics Dept\\
Middle East Technical University\\
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{ex}
    Prove that the following are equivalent:
    \begin{enumerate}
      \item
Every even integer greater than $2$ is the sum of two primes.
\item
Every integer greater than $5$ is the sum of three primes.
    \end{enumerate}
  \end{ex}

  \begin{ex}
    Infinitely many primes are congruent to $-1$ \emph{modulo} $6$.
  \end{ex}

  \begin{ex}
    Find all $n$ such that
    \begin{enumerate}
      \item
$n!$ is square;
\item
$n!+(n+1)!+(n+2)!$ is square.
    \end{enumerate}
  \end{ex}

  \begin{ex}
    Determine whether $a^2\equiv b^2\pmod n\implies a\equiv b\pmod
    n$. 
  \end{ex}

  \begin{ex}
    Compute $\sum_{k=1}^{1001}k^{365}\pmod 5$.
  \end{ex}

  \begin{ex}
    $39\divides 53^{103}+103^{53}$.
  \end{ex}

  \begin{ex}
  Solve  $6^{n+2}+7^{2n+1}\equiv x\pmod{43}$.
  \end{ex}

  \begin{ex}
    Determine whether $a\equiv b\pmod n\implies c^a\equiv c^b\pmod
    n$. 
  \end{ex}

  \begin{ex}
    Determine $r$ such that $a\equiv b\pmod r$ whenever $a\equiv
    b\pmod m$ and $a\equiv b\pmod n$. 
  \end{ex}

  \begin{ex}
    Solve the system
    \begin{equation*}
      \begin{cases}
	x\equiv 1\pmod{17},\\
	x\equiv 8\pmod{19},\\
	x\equiv 16\pmod{21}.
      \end{cases}
    \end{equation*}
  \end{ex}

  \begin{ex}
    The system
    \begin{equation*}
    \begin{cases}
      x\equiv a\mod n\\
      x\equiv b\mod m
    \end{cases}
    \end{equation*}
has a solution if and only if $\gcd(n,m)\divides b-a$.
  \end{ex}

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