Prime Numbers

Chapter 1
PRIMES!
Prime numbers belong to an exclusive world of intellectual conceptions. We
speak of those marvelous notions that enjoy simple, elegant description, yet
lead to extreme—one might say unthinkable—complexity in the details. The
basic notion of primality can be accessible to a child, yet no human mind
harbors anything like a complete picture. In modern times, while theoreticians
continue to grapple with the profundity of the prime numbers, vast toil and
resources have been directed toward the computational aspect, the task of
finding, characterizing, and applying the primes in other domains. It is this
computational aspect on which we concentrate in the ensuing chapters. But we
shall often digress into the theoretical domain in order to illuminate, justify,
and underscore the practical import of the computational algorithms.
Simply put: A prime is a positive integer p having exactly two positive
divisors, namely 1 and p. An integer n is composite if n>1andn is not
prime. (The number 1 is considered neither prime nor composite.) Thus,
an integer n is composite if and only if it admits a nontrivial factorization
n = ab, wherea, b are integers, each strictly between 1 and n. Though the
definition of primality is exquisitely simple, the resulting sequence 2, 3, 5, 7,...
of primes will be the highly nontrivial collective object of our attention. The
wonderful properties, known results, and open conjectures pertaining to the
primes are manifold. We shall cover some of what we believe to be theoretically
interesting, aesthetic, and practical aspects of the primes. Along the way,
we also address the essential problem of factorization of composites, a field
inextricably entwined with the study of the primes themselves.
In the remainder of this chapter we shall introduce our cast of characters,
the primes themselves, and some of the lore that surrounds them.
1.1 Problems and progress
1.1.1 Fundamental theorem and fundamental problem
The primes are the multiplicative building blocks of the natural numbers, as
is seen in the following theorem.
Theorem 1.1.1 (Fundamental theorem of arithmetic). For each natural
number n there is a unique factorization
n = p a1
1 pa2
2 ···pak
k ,