Linear Diophantine Equations

Abstract

We shall describe elementary methods for solving linear Diophantine equations, either single equations or systems of equations. The equations with which we shall deal have integral coefficients and require integral solutions. Such equations are named after Diophantus of Alexandria, who lived probably between 100 and 300 A. D. The methods are explained with the help of specific examples. We begin with the case of a single equation. Method I If an equation has an unknown with coefficient 1 or -1, then all the other unknowns can be taken as arbitrary integers, and the particular unknown is then'determined from the given equation. Exeample 1. 4x + 7y -'z +3w = 18. The solution is: Take x, y. w as arbitrary integers, and determine z from the given equation. It is clear that all solutions in integers are thus obtained. Method JI If the numerically smallest coefficient, say a, is notlor -1, but is a factor of each of the remaining coefficients, and is not a factor of'the right member of the equation (the constantterm), the equation is incompatible, that is, it has no solution in integers. If a divides all the coefficients and also the right member, we divide through by a, thus obtaining an equivalent equation to which Method I can be applied. Exam'ple 2. 8x + 14y2z + 6w = 36. If we divide by 2 we get the equation of Example 1 and can apply Method I. If the right hand side were 37 instead of 36, the equation would have no solution in integers, as 37 is not divisible by 2. Method II can also be applied to any single equation of a system of simultaneous equations. If one equation is thereby shown to be incompatible in integers, then the system is incompatible in integers. We now continue with the case of a single equation. Method III. If the numerically smallest coefficient, say a, is not a factor of all of the remaining coefficients, the equation can be replaced by a new equation with a non-zero coefficient numerically smaller than a, by the method used in the following example. Example 3. 16x+42y-21z2 100. (1) We first set