Overview

Number theory—the study of properties of the positive integers—is one of the oldest branches of mathematics. It has fascinated both amateurs and mathematicians throughout the ages. The subject is tangible, and a great many of its problems are simple to state yet very difficult to solve. "It is just this," said the great nineteenth-century mathematician Carl Friedrich Gauss (1777-1855), "which gives number theory that magical charm which has made it the favorite science of the greatest mathematicians." Indeed, Gauss himself made seminal contributions to the subject, as did such other nineteenth-century greats as Lejeune Dirichlet (1805-1859), Ernst Kummer (1810-1893), Richard Dedekind (1831-1916), Bernhard Riemann (1826-1866), and Leopold Kronecker (1823-1891). Moreover, since the number-theoretic problems they tackled were very difficult, they often had to resort to "nonelementary" means—mainly algebraic...