Number Theory for Computing

Mathematicians do not study objects, but relations among objectsj they are indifferent to the replacement of objects by others as long as relations do not change. Matter is not important, only form interests them. HENRI POINCARE (1854-1912) Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory. IAN STEWART [219] The theory of numbers, in mathematics, is primarily the theory of the prop- erties of integers (i.e., the whole numbers), particularly the positive integers. For example, Euclid proved 2000 years aga in his Elements that there exist infinitely many prime numbers. The subject has long been considered as the purest branch of mathematics, with very few applications to other areas. How- ever, recent years have seen considerable increase in interest in several central topics of number theory, precisely because of their importance and applica- tions in other areas, particularly in computing and information technology.