The subject of this chapter is number theory—the study of the integers and related structures—a subject born in prehistory, already flourishing in ancient Babylon, and still flourishing today; a subject filled with results that are beautiful and magical, and with haunting mysteries; a subject once viewed as the purest of pure mathematics—art for art’s sake—but that in recent years has been applied to problems associated with the transmission, coding, and manipulation of numerical data. (Pure mathematics has become applied mathematics!) Number theory is unique in its abundance of appealing problems whose statements are accessible to elementary school students, but whose solutions have eluded mathematicians for centuries. The struggle to solve problems in number theory led to the development of much of modern algebra, which in turn is linked to an enormous range of subjects in mathematics and the other sciences. Our brief treatment here will be an introduction to some of the most basic properties of the integers. The fundamentals of the divisibility and congruence relations will be explored. A remarkable function, Eiuer’s φ-function, will be introduced, and we will use it to consider this question: Is it possible to show that a given integer is not prime without actually factoring it?
KeywordsNumber Theory Prime Number Identity Element Fundamental Theorem Prime Divisor
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