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\centerline{\bf THEORY OF NUMBERS, Math 115 B}
\centerline{\bf Homework 1 Due Friday January 17}
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\item For each $\epsilon>0$ below find lattice points $(p,q)$ whose
distance from the line $y=\sqrt{2}x$ is less than $\epsilon$:

a) $\epsilon=1/2$, b) $\epsilon=1/5$, c) $\epsilon=1/10$ 

\item Are there infinite strips between parallel lines that are
completely lattice-point free?

\item Prove that every triangle with three non-collinear vertices
at lattice points, but no other lattice points on its boundary and
no interior lattice points, has an area of $1/2$.
(HINT: Consider, as in class, the smallest rectangle that contains
the triangle and see whether you can write a formula for the area
in terms of rightangle triangles and quadrilaterals).


\item Prove that for any triangulation of a polygon without holes.
What is the parity of the sum $V+E+F$? $V,E,F$ denote the number of vertices, edges,and triangles in the triangulation.

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