The volume is devoted to the interaction of modern scientific computation and classical number theory. The contributions, ranging from effective finiteness results to efficient algorithms in elementary, analytical and algebraic number theory, provide a broad view of the methods and results encountered in the new and rapidly developing area of computational number theory. Topics covered include finite fields, quadratic forms, number fields, modular forms, elliptic curves and diophantine equations. In addition, two new number theoretical software packages, KANT and SIMATH, are described in detail with emphasis on algorithms in algebraic number theory.

This book constitutes the refereed proceedings of the 7th International Algorithmic Number Theory Symposium, ANTS 2006, held in Berlin, July 2006. The book presents 37 revised full papers together with 4 invited papers selected for inclusion. The papers are organized in topical sections on algebraic number theory, analytic and elementary number theory, lattices, curves and varieties over fields of characteristic zero, curves over finite fields and applications, and discrete logarithms.

Themen sind die grundlegenden arithmetischen und algebraischen Objekte: ganze Zahlen, endliche Korper, euklidische Ringe und Polynomringe. Es behandelt Algorithmen fur Primzahltests, Faktorisierungsmethoden fur ganze Zahlen und Polynome sowie Verfahren zur Berechnung von Grobner Basen. Besondere Aufmerksamkeit wird der Darstellung der behandelten Objekte, der Analyse der Algorithmen und der Losung diophantischer Gleichungen und Gleichungssysteme gewidmet.