Number Theory for the Millennium III

Description

Building on the tradition of an outstanding series of conferences at the University of Illinois at Urbana-Champaign, the organizers attracted an international group of scholars to open the new Millennium with a conference that reviewed the current state of number theory research and pointed to future directions in the field. The conference was the largest general number theory conference in recent history, featuring a total of 159 talks, with the plenary lectures given by George Andrews, Jean Bourgain, Kevin Ford, Ron Graham, Andrew Granville, Roger Heath-Brown, Christopher Hooley, Winnie Li, Kumar Murty, Mel Nathanson, Ken Ono, Carl Pomerance, Bjorn Poonen, Wolfgang Schmidt, Chris Skinner, K. Soundararajan, Robert Tijdeman, Robert Vaughan, and Hugh Williams. The Proceedings Volumes of the conference review some of the major number theory achievements of this century and to chart some of the directions in which the subject will be heading during the new century. These volumes will serve as a useful reference to researchers in the area and an introduction to topics of current interest in number theory for a general audience in mathematics. Cover Half Title Title Page Copyright Page Table of Contents Preface List of Contributors Volume 3 Some Remarks on Primality Testing Based on Lucas Functions Splitting of Primes in Infinite Extensions The ABC Conjecture and Prime Divisors of the Lucas and Lehmer Sequences On the Parity of Generalized Partition Functions Quadratic Twists of Modular Forms and Elliptic Curves Identities from the Holomorphic Projection of Modular Forms Iterative Methods for Pairs of Additive Diophantine Equations Quelques Remarques Sur La Théorie D’Iwasawa Des Courbes Elliptiques Computing Rational Points on Curves Norms of Products and Factors of Polynomials G. H. Hardy As I Knew Him Discriminants of Some Painlevé Polynomials Identities Between Mahler Measures Normal Integral Bases, Swan Modules and p-Adic L-Functions Discriminants and Divisibility for Chebyshev-Like Polynomials Remarks on Adelic Geometry of Numbers Some Applications of Diophantine Approximation On the Solutions of a Family of Sextic Thue Equations Waring’s Problem: A Survey Chernoff Type Bounds for Sums of Dependent Random Variables and Applications in Additive Number Theory Prime Divisors of the Bernoulli and Euler Numbers An Improved Method for Solving the Family of Thue Equations X4 – 2rX2Y2 – sY4 = 1 Constructing Hyperelliptic Curves Using Complex Multiplication Solving the Pell Equation A Central Limit Theorem for the Number of Distinct Degrees of Prime Factors in Additive Arithmetical Semigroups