Fundamentals of Number Theory

Abstract

Throughout this book we shall mainly work with integers unless we specify otherwise. By N we denote the set {1, 2, 3, ... } of all natural numbers, i.e., all positive integers. A natural number p is said to be prime if p > 1 and p is divisible only by p and 1. Thus, the natural numbers are divided into the unit 1, prime numbers, and composite numbers. Notice that we can find arbitrarily many successive natural numbers that are all composite. Indeed, for any n > 1 the sequence n! + 2, n! + 3, ... , n! + n contains n - 1 successive integers that are evidently composite. On the other hand we have the following theorem of Euclid:

• Theorem 2.1(Euclid) The number of primes is infinite.

The distribution of primitive roots is a deep mystery

Carl Friedrich Gauss