Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

3. Fundamentals from Number Theory
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.
3.1 Divisibility and Greatest Common Divisor
The actual object of our studies in this book are the natural numbers
0, 1, 2, 3,...and the prime numbers 2, 3, 5, 7, 11,.... Still, it is necessary to
start with considering the set
Z = {0, 1, −1, 2, −2, 3, −3,...}
of integers together with the standard operations of addition, subtraction,
and multiplication. This structure is algebraically much more convenient than
the natural numbers.
Definition 3.1.1. An integer n divides an integer m, insymbolsn | m, if
nx = m for some integer x. If this is the case we also say that n is a divisor
of m or that m is a multiple of n.
Example 3.1.2. Every integer is a multiple of 1 and of −1; the multiples
of 3 are 0, 3, −3, 6, −6, 9, −9, .... The number 0 does not divide anything
excepting 0 itself. On the other hand, 0 is a multiple of every integer.
We list some elementary properties of the divisibility relation, which will
be used without further comment later.
M. Dietzfelbinger: Primality Testing in Polynomial Time, LNCS 3000, pp. 23-53, 2004.
© Springer-Verlag Berlin Heidelberg 2004