THEORY OF NUMBERS, Math 115 B Homework

1. Prove that any triangulation of a simple polygon with n vertices has n − 3 diagonals and n − 2 triangles.

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

3. 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).

4. Determine the number of lattice points in the segments (12, 24) − (17, 20) (3/4, 8/7) − (3, 11/7)

5. 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).

6. 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.

7. 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.

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

9. (application of Minkowski theorem) Using Minkowski’s geometric thinking prove the following number theory statement Given any real number α and an integer t > 0, there exist integers p, q no both zero such that |q/p − α| ≤ 1/t. (HINT: take the right M-set and apply Minkowski’s theorem to it).