On The Solutions Of The Diophantine Equations X～3±p～3=Dy～2

The diophantine equation is the oldest branch in number theory, whose content is extremely abundant, and it has close connections with the algebraic number theory, the algebraic geometry, the combinatorics and so on. In the recent 30 years, this field also has developed too much. In such fields as the information encoding theory, the algebraic number theory and the diophantine analysis theory, many types of the result of cubic diophantine equation are used, which make it necessary for us to study some basic types of the solutions of cubic diophantine equation. We are familiar to the solution of the simple diophantine equation and quadratic diophantine equation, while with the solution of cubic diophantine equation, there is no general conclusion, so it needs further discussing. The experts of Number theory, such as Ke Zhao, Sun Qi, have studied much about the following equation: x 3±( p k)3 = Dy2 (where D doesn't contain prime factor of the form 6 k +1), and drawn the general conclusion. But as for the situation of only containing one prime factor of the form 6 k +1, few people have been studying.In this paper,I discussed the integer solutions of the Diophantine equation x 3±p3=Dy2 with the elementary method,in which D is a square– free integer and only contain one prime factor of the form 6 k +1 ( D >0), and p is an arbitrary prime, some sufficient conditions are given when the equations have no integer solutions.The whole paper is divided into three parts, and its main content is as follows:...