SOLVING THE PELL EQUATION BY THE LAGRANGE'S THEOREM METHOD

Abstract

The terms and expressions in the theory of numbers , at a superficial observation may seem so simple and useless in practice. However , if you look at them based on their actual positions of their historical genesis and evo- lution, which are essentially caused by the general social practices and the requirements of mathematics as a science, having in mind that these are the indirect expression of the needs and requirements of the general social practice, then the situation changes in terms of these concepts and statements and the way we see it, because they arise from the constant collision between the reality and the man, without which they would have been concealed in the man only as a possibility.

This is why in the theory of numbers, both in mathematics and every other science, it is very important to consider the "present" in connection with "the past", because the "present" developed from "the past", and also in the same way "the future" will develop from "the present". The study of "the past" in every science, even in mathema- tics, and thus also in the theory of numbers, becomes a means to understand "the present" and predict "the future". Therefore, this conceives the development of each of the mathematical models and its complete theory as a part of historical process within the general framework of the development of social practice, whether they are directly or indirectly related to it. This clearly causes the basic problems to appear along with all the tasks and objectives of the research that are normally dealt within the field of research of the history of mathematics.

The subject of the research is to solve the Pell's equation. In order to illustrate the possible size of the num- bers that appear in resolving the Pell's equation, it is necessary to describe "the Archimedes' cattle problem." The properties of continued fractions have also been given, due to their application and use in solving the Pell's equation by using the Lagrange equations .