Pell Equation Theory In A Class Of Diophantine Equations

Diophantine equation, which is called indeterminate equation, is the equation, in which the number of the variables that are integer is more than the number of the equations, and it is a branch of number theory and has a wide range of research value. It is closely linked with the Math, the algebraic geometry and the algebraic coding. Any of its results not only plays a catalytic role in all branches of mathematics but also is of great practical significance towards the non-mathematical subjects, such as the Computer Sciences, Electronics and Digital Signal Processing. There are many methods to solve the Diophantine equation, including the elementary methods and the advanced methods. But there is a fact that there is not a general algorithm for all the Diophantine equation to judge whether it has a solution, which brings great difficulties to us. Pell equation, which is as the most ancient Diophantine equation, has aroused great concern for the scholars of the Number Theory. There are monumental achievements to the study of the Pell equation. Pell equation theory is greatly useful for solving a class of Diophantine equation. The main work of this paper is as follows:1,The issue of minimum solution of the Pell equation x2-Dy2=±1 with the elementary methods was researched and solving a number of specific ways of the Diophantine equation were given2,The issue of the existence of the solution for a class of Diophantine equations with the relevant theories of the Pell equations is proposed...