On The Diophantine Equation X～3±8=Dy～2

The Diophantine equation not only develop activitily, but also be apply to Discrete Mathematics else field, It plays a key role in people′s study and research and solving the actual problems. So many researchers study to the extensive and higher the Diophantine equation in the internal and the external.Several authors have studied on the Diophantine equation x~3±8= Dy~2. For D≠6k+1, all positive integer solution of have been obtained. When D has not square factor and has prime factor≡1 ( mod6), it has difficulty. The equation x~3±8= Dy~2 (0﹤D﹤50),When D = 13,21,31,35,37,39,43 have not been solution.In this paper, with the method of recurrence sequences and maple formality and Pell equation and quadratic residue. we have shown seven Diophantine equation the only solution in positive integers. The first part, we give x~3±8= Dy~2for the condition and the significance ; The second part, we have proved the Diophantine equation x~3 + 8= Dy~2,When D = 13,21,31,37,39has only integer solution (x,y)=(－2,0);x~3－8= Dy~2,When D = 21,31,35,37,39,43 has only integer solution (x,y) = (2,0); x~3－8 =13y~2has only integer solutions (x,y) = (－2,0), (5,±3) (6,±4) , (626,±4344); x~3 + 8 =35y~2 has only integer solutions (x,y)=(－2,0),(3,±1); x~3 + 8 = 43y~2 has only integer solutions (x,y) = (－2,0), (14,±8).The 4-th part, 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 gather in the third part.