On The Diophantine Equation X～2+D=4y～5

The Diophantine equation not only developed actively itself, but also was apply to else fields of Discrete Mathematics. It plays an important role in people's study and research to solve the actual problems. 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 Theory obtained the first formation and developments. Currently, Algebraic Number Theory has become a branch of mathematics with abundant contents, is also an important tool of studying of the Diophantine equation.As an important part of Algebraic Number Theory, quadratic fields and the arithmetic in quadratic fields play an important role in research of the Diophantine equation. For some D , Q ( D ) is a Euclid area, And arithmetical theorem is carried out over the rings of quadratic integers. We can build up the same division theorem to solve some Diophantine equation. There are several work of study of solving the Diophantine equation x~2 + D = 4 pn with knowledge of Algebraic Number Theory, especially the study of the Diophantine equation x~2 + D = 4 pn according to discuss the divisibility of class number in imaginary quadratic fields. But it's difficult to solve it in actual quadratic fields.In this paper, we will prove all the integer solutions of the Diophantine equation x~2 + D = 4y~5( D =7,11,-5,-13,-21,-29)with the method of some important theories in quadratic fields, the rings of quadratic integers Q ( D ) and Maple program in four parts. In the first part, we give the present condition of the Diophantine equation x~2 + D = 4 pn in the domestic and abroad. In the second part, we give the preparation knowledge of the whole paper. We introduced important theories in quadratic fields, quadratic Euclid area and the arithmetical theorem in the rings of quadratic integers Q ( D ) in detail. In the third part, we well prove that the Diophantine equation x~2 + 7 = 4y~5 has integer solutions ( x , y )=(±11,2);the Diophantine equation x~2 + 11 = 4y~5 has integer solutions ( x , y )=(±31,3);the Diophantine equation x~2 - 5 = 4y~5 has integer solutions ( x , y )=(±1,-1)and (±3,1);the Diophantine equation x~2 - 13 = 4y~5 has integer solutions ( x , y )=(±3, -1);the Diophantine equation x~2 - 21 = 4y~5 has integer solutions ( x , y )=(±5 ,1);the Diophantine equation x~2 - 29 = 4y~5 has integer solutions ( x , y )=(±5, -1).In the fourth part we summarize the whole paper, 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.