# An Algebraic Treatment of Diophantine Analysis

5-9-1940

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

Mathematics

W.V. Parker

## Abstract

Diophantlne analysis, while an old subject, is a collection of special problems. The methods used are very varied and those used to solve one problem will not usually solve a second one. It is possible that this condition exists because Diophantlne analysis is usually considered as number theory. While some problems are problems in number theory, a large body of material may be developed by algebraic methods. In this thesis the methods of algebra are exploited. The material of this thesis divides itself into chapters naturally, according to the degree of the equations. Considered in order are quadratics, cubics, quartics, and equations of degree n. All equations considered are homogeneous polynomials except a few of degree n. All except two equations are generalized to equations of degree n. The solutions are given in terms of arbitrary parameters and are Integral for an integral choice of the parameters. An important concept of this thesis is the introduction of equivalent solutions (due to Professor W. V. Parker). Consider the Diophantlne equation U) f(x, ,...,xf) = g(y, ,...»y\$)> ® where f and g are homogeneous polynomials with integral coefficients , of degrees m and n respectively. If xt = ^ , yy * fiy is a solution of (1) and there are no integers s > 1, (5*5) o('» fa such that rm *n, then X< = , yis defined to be a primitive solution of (1). If x4 = < , yy = ft is a primitive solution of (1), then xt =. t* , yy ■= fc , where t is a non-zero integer, is also a solution which is said to be derived from this primitive (V) f • solution. Two solutions are defined to be equivalent if they may be derived from the same primitive solution. Eaoh solution obtained is shown, within certain limits, to be general. For a given solution of one of the equations considered, the parameters are chosen in such a way that a solution, equivalent to the given one, is obtained, provided the given solution is not also a solution of a certain equation. Usually, if the original equation is of degree n, the "certain" equation is of degree n - 1.