Introduction and Historical Remarks

Abstract

The theory of numbers is concerned with the properties of the integers, that is, the class of whole numbers and zero, 0,±1,±2 .... The positive integers, 1, 2, 3 ..., are called the natural numbers. The basic additive structure of the integers is relatively simple. Mathematically it is just an infinite cyclic group (see Chapter 2). Therefore the true interest lies in the multiplicative structure and the interplay between the additive and multiplicative structures. Given the simplicity of the additive structure, one of the enduring fascinations of the theory of numbers is that there are so many easily stated and easily understood problems and results whose proofs are either unknown or incredibly difficult. Perhaps the most famous of these was Fermat’s big theorem, which was stated about 1650 and only recently proved by A. Wiles. This result said that the equation aⁿ+bⁿ = cⁿ has no nontrivial (abc ≠ 0) integral solutions if n > 2. Wiles’s proof ultimately involved the very deep theory of elliptic curves. Another result in this category is the Goldbach conjecture, first given about 1740 and still open. This states that any even integer greater than 2 is the sum of two primes. Another of the fascinations of number theory is that many results seem almost magical. The prime number theorem, which describes the asymptotic distribution of the prime numbers has often been touted as the most surprising result in mathematics.