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\title{Number-theory exercises, III}
\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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Here $p$ and $p_i$ are always prime numbers.

  \begin{ex}
    $p\equiv\pm1\pmod6$.
  \end{ex}

  \begin{ex}
    If $p\equiv1\pmod3$ then $p\equiv1\pmod6$.
  \end{ex}

  \begin{ex}
    If $n\equiv2\pmod3$, then $n$ has a factor $p$ such that
    $p\equiv2\pmod3$. 
  \end{ex}

  \begin{ex}
    Find all primes of the form $n^3-1$.
  \end{ex}

  \begin{ex}
    Find all $p$ such that $3p+1$ is square.
  \end{ex}

  \begin{ex}
    Find all $p$ such that $p^2+2$ is prime.
  \end{ex}

  \begin{ex}
    $n^4+4$ is composite.
  \end{ex}

  \begin{ex}
    If $n$ is positive, then $8^n+1$ is composite.
  \end{ex}

  \begin{ex}
    Find all integers $n$ such that the equation
    \begin{equation*}
      x^2=ny^2
    \end{equation*}
has only the zero solution.  Prove your findings.
  \end{ex}

  \begin{ex}
    If $p_0<\dotsb<p_n$, prove that the sum
    \begin{equation*}
      \frac1{p_0}+\dotsb+\frac1{p_n}
    \end{equation*}
is not an integer.
  \end{ex}

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