The Solution of Equations in Integers

INTRODUCTION

The theory of numbers covers the basic arithmetical properties of the natural number series, that is, the positive integers, and belongs to the oldest branch of mathematics. One of the central problems of the so-called analytic theory of numbers is concerned with the distribution of the prime numbers in the natural number series. A prime number is any positive integer greater than unity and divisible without remainder only by itself and unity. The problem of the distribution of prime numbers in the natural number series consists in determining the regularity of the behavior of the number of prime numbers less than for large values of . The first result in this direction we find in Euclid (400 B.C.), namely, a proof of the infinitude of the series of prime numbers; the second result was obtained by the famous Russian mathematician P. L. Cheby-cheff in the second half of the nineteenth century. Another basic problem in the theory of numbers is the problem of the representation of integers as the sum of integers of specified type—for example, representing odd integers as the sum of three prime integers. The last problem, the problem of Goldbach, was solved* comparatively recently by the most significant present-day representative of the theory of numbers—the Soviet mathematician I. M. Vinogradoff.

This book is dedicated to a very interesting branch of the theory of numbers, the solution of equations in integers. The determination of integral solutions of algebraic equations with integral coefficients and with more than one unknown is one of the most difficult problems in number theory. Many of the outstanding mathematicians of antiquity were concerned with this problem—the Greek mathematician Pythagoras (sixth century B.C.), the Alexandrian mathematician Diophantus (third century B.C.)—as well as some of the best mathematicians nearer our time—P. Fermat (seventeenth century), L. Euler and J. Lagrange (eighteenth century), and others. Nevertheless, the efforts of many generations of outstanding mathematicians have failed to yield any general methods comparable to the Vinogradoff method of trigonometric sums which has been used to solve quite distinct problems in the analytic theory of numbers.

The problem of solving equations in integers is completely settled only for equations of second degree in two unknowns. Equations of any degree with one unknown do not present any interest since they can be solved with the help of a finite number of trials. For equations of degree higher than the second with two or more unknowns, it is a very difficult problem not only to find all solutions in integers, but even to answer the simpler question whether a finite or an infinite number of such solutions exists.

Since equations in integers are encountered in physics, the solution of these equations is of more than theoretical interest. However, the theoretical interest in equations in integers is very great since these equations are closely connected with many problems in the theory of numbers. In addition, the elementary parts of the theory of such equations, which are discussed in this book, may be used to widen the horizons both of students and teachers.

In this book are expounded certain fundamental results obtained in the theory of solving equations in integers. The theorms formulated are proved when these proofs are not too difficult.

* Not strictly correct since what Vinogradoff showed was only that every sufficiently large integer can be written as the sum of three primes [Translator].

1. Equations in One Unknown

Let us consider equations of the first degree in one unknown

Let the coefficients a1 and a0 in the equation be integers. It is clear that the solution of this equation,

will be an integer only if a1 integrally divides a0. Thus equation (1) is not always solvable in integers. For example, of the two equations 3x – 27 = 0 and 5x + 21 = 0, the first has the integer solution x = 9, but the second is unsolvable in integers.

The same situation is met in equations of degree higher than the first. The quadratic equation x² + x – 2 = 0 has integer solutions x1 = 1, x2 = –2; but the equation x² + 4x + 2 =