More Number Theory

Abstract

We have already discussed some concepts in the theory of numbers in Chapter 4, in our study of the Euclidean algorithm, continued fractions, and Fibonacci numbers. Gauss’s stirring quotation above encourages us to pursue this topic further. In this chapter we begin with the glorious theorem from ancient Greek mathematics that so elegantly demonstrates that the number of primes is infinite. We will discuss other properties of prime numbers, including how irregular they are “in the small,” yet how orderly they are “in the large.” The concept of congruences, developed by Gauss, will be used to obtain results concerning divisibility. The theory of quadratic residues leads us on to Wilson’s theorem, Gauss’s lemma, and Gauss’s law of quadratic reciprocity. Much of this chapter is devoted to the study of Diophantine equations, of which solutions are sought in integers. This is the area in which the notorious Fermat’s last theorem lies, that there are no solutions in positive integers x, y, and z of the equation x ⁿ + y ⁿ = z ⁿ if n is an integer greater than 2. A study of Andrew Wiles’s proof of this, published in 1995, is very much beyond the scope of this book. However, we give proofs for the special cases where n = 3 and n = 4. As a prelude to the proof of the case where n = 3, we first discuss properties of algebraic integers, which is a fascinating topic in its own right. The reader may agree that the successful ascent of the subproblem of Fermat’s last theorem when n = 3 is sufficient cause for celebration.

Mathematics is the queen of the sciences, and tha theory of numbers is the queen of mathematics.