About Diophantine Equation X~3±27=Dy~2

The Diophantine equation is an old and interesting subject in number the-ory, It involves many details. It is greatly connected with algebraic number theory, algebraic geometry and combinatorics etc. It plays an important role in solving practical problems. Therefore, many external and internal researchers do the extensive and intensive study on the Diophantine equation.Several authors have studied on the Diophantine equation x3±27= Dy2(D> 0). For D has no prime factor?1(mod6), all positive integer solutions of them have been obtained. When D has no square factor and has prime fac-tor?1(mod6),it has difficulty. When D=14,31,35,37,38,43, the Diophantine equation x3±27=Dy2(0<D< 50)have not been solved.In this paper, with the method of recurrence sequences and congruence equation and Pell equation and quadratic residue and Matlab formality, we have researched the integer solutions of the Diophantine equation x3±27= Dy2(D={14,31,35,37,38,43}). The first chapter, we introduce the external and internal condition and significance of x3 ±27=Dy2; The second chap-ter, we give all the basic knowledge; The third chapter is the priority of the paper. In this part, we give all the integer solutions of x3±27=Dy2(D= {14,31,35,37,38,43});In the forth chapter, we summarize the total paper, and put forward some problems which should be solved in perhaps development direction in the future. In this paper, main result will be gathered in the third part.