Diophantine Olympics and World Champions: Polynomials and Primes Down Under

1. LET THE GAMES BEGIN: THE OPENING CEREMONIES. For those who think globally, "down under" may provoke thoughts of Australia-the home of the 2000 Olympic Games. For those who think rationally, "down under" may provoke thoughts of denominators of fractions. In this paper, we hope to provoke both. In basic diophantine approximation, the name of the game is to tackle the following: How close do integer multiples of an irrational number get to integral values? Specifically, if a is an irrational number and the function 11 11 on OR gives the distance to the nearest integer (that is, lixll = mintlx ml: m E E}), then the game really is a competition among all integers n to minimize the value II an II. Suppose someone serves us an irrational number a. We write ql, q2, q3, ... for the (winning) sequence of integers that is able to make it over the following three hurdles: