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\centerline{\bf THEORY OF NUMBERS, Math 115 B}
\centerline{\bf Homework }
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\begin{enumerate}

\item Prove that any triangulation of a simple polygon with $n$ vertices
has $n-3$ diagonals and $n-2$ triangles.

\item Prove that any triangulation of a simple polygon has an ear:
i.e. a triangle that shares a single edge

\item Suppose $a,b$ are relatively prime integers. Prove
that there are no lattice points in the interior of the
segment going from $(0,0)$ to $(a,b)$.

\item Determine the number of lattice points in the segments
$(12,24)-(17,20)$
$(3/4,8/7)-(3,11/7)$

\item Find out the number of lattice points inside the polygon
$P=v_0,v_1,...v_5$ with vertices $v_0=(0,0)$, $v_1=(1,1)$, $v_2=(2,4)$,
$v_3=(3,9)$, $v_4=(4,16)$, $v_5=(5,25)$. 

\item Using straighline segments, draw a nonconvex figure with
symmetry about $(0,0)$, area more than 4, and no lattice points
except $(0,0)$ inside it or on its boundary.

\item Find a tetrahedron with integer coordinates for its vertices
but no lattice points on its boundary or interior, whose volume is
more than $1/6$. This makes a generalization of Pick's theorem
impossible.

\item A convex set $Q$ contains three non-collinear points $A,B,C$. Prove that
then $Q$ must contain every point of the triangle $ABC$.

\item (application of Minkowski theorem) Using Minkowski's geometric 
thinking prove the following number theory statement
Given any real number $\alpha$ and an integer $t&gt;0$, there exist integers
$p,q$ no both zero such that $\vert q/p-\alpha \vert \leq 1/t$. (HINT:
take the right M-set and apply Minkowski's theorem to it).

\end{enumerate}

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