Giuliana P. Davidoff

Elementary Number Theory, Group Theory, and Ramanujan Graphs: An Overview

An overview 1. Graph theory 2. Number theory 3. PSL2(q) 4. The graphs Xp,q Appendix A. 4-regular graphs with large girth Index Bibliography.

The Geometry of Numbers: Solutions and Hints

Chapter 1 1 a. y = ⅔ x + ⅕ (because 5 does not divide 3). Other examples are similar. b. (because 3 | 15). ( p 0 , q 0 ) = (5, 1); {( p k , q k ) = (5 + 15 k , 1 + 2 k ), | k is any integer}. 2. Hint: Use the distance formula to find the length d k of the line segment from ( p k -1 , q k -1 ) to ( p k , q k ). If d k is not a function of k , then d k is constant from any point to any adjacent lattice point.