3. Fundamentals from Number Theory

Abstract

In this chapter, we study those notions from number theory that are essential for divisibility problems and for the primality problem. It is important to understand the notion of greatest common divisors and the Euclidean Algorithm, which calculates greatest common divisors, and its variants. The Euclidean Algorithm is the epitome of efficiency among the number-theoretical algorithms. Further, we introduce modular arithmetic, which is basic for all primality tests considered here. The Chinese Remainder Theorem is a basic tool in the analysis of the randomized primality tests. Some important properties of prime numbers are studied, in particular the unique factorization theorem. Finally, both as a general background and as a basis for the analysis of the deterministic primality test, some theorems on the density of prime numbers in the natural numbers are proved.