ABSTRACT
The study of various properties of prime numbers is not only themost important content in number theory, but it also plays an important role in cryptography. In previous chapters, we have had discussions of some of the properties of prime numbers. This chapter will explore some other properties and the problem about how primes are distributed in the set of positive integers by describing the prime number theorem. We also introduce an important result for the prime distribution in an arithmetic progression-the prime number theorem in arithmetic progressions. The results of this chapter have many applications in cryptography. For example, the general prime number theorem and the prime number theorem in an arithmetic progression explain the average probabilities of prime distribution in the set of positive integers and in an arithmetic progression, respectively. These provide the theoretical basis for probabilistic algorithms of generating prime numbers.
