THEORY OF NUMBERS, Math 115 B Homework 1 Due Friday January 17

1. For each ϵ > 0 below find lattice points (p, q) whose distance from the line y = \sqrt{2}x is less than ϵ:

a) ϵ = 1/2, b) ϵ = 1/5, c) ϵ = 1/10

2. Are there infinite strips between parallel lines that are completely latticepoint free?

3. 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).

4. 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.