Computational Number Theory

Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented geometry. To help predict the positions of the planets, the Greeks invented trigonometry. Algebra was invented to deal with equations that arose when mathematics was used to model the world. The list goes on, and it is not just historical. If anything, computation is more important than ever. Much of modern technology rests on algorithms that compute quickly: examples range from the wavelets that allow CAT scans, to the numerical extrapolation of extremely complex systems in order to predict weather and global warming, and to the combinatorial algorithms that lie behind Internet search engines (see Section ?? of The Mathematics of Algorithm Design). In pure mathematics we also compute, and many of our great theorems and conjectures are, at root, motivated by computational experience. It is said that Gauss, who was an excellent computationalist, needed only to work out a concrete example or two to discover, and then prove, the underlying theorem. While some branches of pure mathematics have perhaps lost contact with their computational origins, the advent of cheap computational power and convenient mathematical software has helped to reverse this trend. One mathematical area where the new emphasis on computation can be clearly felt is number theory, and that is the main topic of this article. A prescient callto-arms was issued by Gauss as long ago as 1801: