On The Diophantine Equation X~2+D=4y~7

The Diophantine Equations not only developed actively itself, but also were applied to other fields of Discrete Mathematics. They play an important role in people'study and research .So many researchers study the Diophantine Equation extensively and highly in the domestic and abroad. Along with the development of the Diophantine Equation, Algebraic Number obtained the primary formation and development. Especial after the mathematician Kummer introduced the concept of ideal, the study of Diophantine Equations has a great breakthrough.As an important part of Algebraic Number, quadratic fields ,the arithmetic in quadratic fields ,ideal ,class number and unique factorization theory play an important role in research of Diophantine Equation. For some D , as for the integer solution of Diophantine Equation x² + D=4y⁷, if the class number of Q(D^1/2) is 1, we will directly apply its corresponding unique factorization theory in algebraic integer domain , if the class number of quadratic fields Q(D^1/2) is not 1, we will first apply its corresponding unique factorization theory of ideal in algebraic integer domain ,then apply the property of principal ideal domain being unique factorization domain to solve.In this paper , we will prove all the integer solutions of the Diophantine Equation. x² + D=4 y⁷( D >0) with the method of some important theories in quadratic fields and Maple program in four parts. In the first part, we give the present condition of Diophantine Equation x² + D=4pnin the domestic and abroad. In the second part, we give the preparation knowledge of the whole paper: we introduce important properties in quadratic fields, ideal, class number and unique factorization theory in algebraic integer domain and so on. In the third part, we prove integer solutions of Diophantine equations x² + D=4y⁷( D >0) in three sections. In section I, we prove that the Diophantine equation x² + 3 =4y⁷and x² + 19 =4y⁷ have only integer solutions ( x ,y)=(±1,1)and ( x ,y)=(±559,5). In section II, we prove that the Diophantine equation x² + D=4y⁷( D =7,11,43,67)have no integer solutions. In section III , we prove that the Diophantine equation x² + D=4y⁷( D =23,31,47,59) have no integer solutions .In the fourth part, We summarize the whole paper, sum up the general situation and put forward some problems which should be solved in perhaps development direction in the future. In this paper , main result will be given in the third part.