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\begin{center}
  {\large\textbf{Dartmouth College}\\
    Mathematics 25}
\end{center}
\begin{center}
  Assignment 2\\
  due Wednesday, October 7
\end{center}

\begin{enumerate}

\item Find all integer solutions to $11305 x + 455 y = 105$.

\item Let $m,n \in \Z$, and consider the lattice point $(m,n)$ in the
  plane.  We say that the point $(m,n)$ is visible from the origin,
  $(0,0)$, if and only if there are no other lattice points on the
  line segment between $(0,0)$ and $(m,n)$.  Prove that $(m,n)$ is
  visible from the origin if and only if $\gcd(m,n) = 1$.  You may, if
  you find it convenient, assume that $m,n \ge 0$, though this is not
  necessary.

\item Prove that there are infinitely many integers $m$ so that $6m+5$
  is prime.

\item Our general theory tells us that there are infinitely many
  solutions to the Diophantine equation $4x + 5y = n$ for any integer
  $n$.  Let $P(n)$ be the statement that $4x + 5y = n$ is solvable for
  nonnegative $x$ and $y$.  Show that $P(n)$ is true for all $n \ge
  12$.  Induction is one means to this end.

\item The Fibonacci numbers $f_n$ are ubiquitous. They are defined
  recursively by $f_1 = f_2 = 1$, and for $n \ge 3$, $f_n = f_{n-1} +
  f_{n-2}$; thus the sequence begins $1,1,2,3,5,8,13,\dots.$ Show that
  $0\le f_n < f_{n+1}$ for all $n \ge 2$.  For $n \ge 3$, show that
  $\gcd(f_{n+2}, f_{n+1}) = 1$ and that the Euclid's algorithm takes
  exactly $n$ steps to run.

\item On an island live 17 people all of whom have blue eyes. They are
  capable of perfect logic and live by two rules:
  \begin{compactitem}
  \item No person reveals the color of another person's eyes to that
    person.
  \item If a person learns that their eyes are blue, they must jump
    into the island's volcano at midnight of the day they learn this
    fact.
  \end{compactitem}

  A missionary comes to the island with the intent of converting the
  residents to the ``true faith'', but after months of effort fails to
  convert even a single person.  Frustrated, he leaves, but just as he
  departs, the residents all gather at the beach and the missionary
  says, ``at least one of you has blue eyes.''

  \begin{enumerate}
  \item Describe what happens over the next year.  Hint: This is an
    induction problem.

  \item What, if anything, did the missionary tell the residents that
    they didn't already know?
  \end{enumerate}

\end{enumerate}

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