Algebraic Aspects of Number Theory

Abstract

Chapter 10 discusses some more interesting properties of integers, in particular, properties of prime numbers and primality testing by using the tools of modern algebra, which are not studied in Chap. 1. In addition, the applications of number theory, particularly those directed towards theoretical computer science, are presented. Number theory has been used in many ways to devise algorithms for efficient computer and for computer operations with large integers. Both algebra and number theory play together an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to the fields of coding theory and cryptography. The motivation of this chapter is to provide an introduction to the algebraic aspects of number theory, mainly the study of development of the theory of prime numbers with an emphasis on algorithms and applications, necessary for studying cryptography, to be discussed in Chap. 12. This chapter starts with the introduction to prime numbers with a brief history. Several proofs of the celebrated theorem of Euclid stating that there exist infinitely many primes are given in this chapter. Further discussion of Fermat numbers, Mersenne numbers, Carmichael numbers, quadratic reciprocity, multiplicative functions such as Euler phi-function, number of divisor functions, sum of divisor functions etc. has been made. This chapter ends with a study on primality testing both deterministic and probabilistic such as Solovay–Strassen and Miller–Rabin probabilistic primality tests.