A Class Of Higher Order Diophantine Equation Solving

The so-called Diophantine equation is the indeterminate equation in number theory in which the number of variable is more than that of the equation (or equations). Diophantine equation is an important branch in number theory and closely connected with algebraic number theory, algebraic geometry, combinatorics and computer science and so on . Many unsolved and difficult problems are reduced to some indeterminate equations. Rich in its content and no unified approach contribute to the difficulty of its solvability. In general, we can only afford some solving principles which combine the elementary and advanced methods to transform the Diophantine to some equations easy to deal with or with well-known results.We are familiar with the method and solution of the simple Diophantine equation and quadratic Diophantine equation . But for the solution of cubic Diophantine equation and high order Diophantine equations ,there is no general conclusion, so it needs further discussing .The main contents of this paper are:1. It is showed in this paper the concept of Diophantine equation, recent research and approaches to deal with it.2. Prior knowledge to introduce congruence theory and some important theorems in Euclid field and integer ring is presented.3. Some special cases of the general form of the Diophantine equationAx~2 +B = y~n are discussed in next five parts.