On The Diophantine Equations X~3±27=Dy~2

Diophantine equations are not only active in itself, but also are applied to other fields in Discrete Mathematics. They play a key role in study and solving actual problems. Therefore there are many scholars home and abroad who do broadly and deeply research in them.Many scholars have done lots of work on the Diophantine equations x~3±27 =Dy~2( D > 0). Especially when D has no square divisors and has no prime factors in the form of 6 k +1, there are many good results. But when D has no square divisors but has prime factors in the form of 6 k +1, it is difficult to solve the equations. As 3| x,0 < 3D<100 , the Diophantine equations x~3±27 =Dy~2 can transform into x1 3±1 = 3Dy12, the integer solutions have been solved; but as 3 /| x, 0 < 3D<100 , the integer solutions of x~3±27 =Dy~2 have not been solved yet.In this paper, with the methods of recurrence sequences, congruence, maple program, the characters of Pell equations and quadratic residue, we prove all the integer solutions of the five group Diophantine equations x~3±27 =Dy~2( D=7 ,13 ,19 , 26 , 31).In the first chapter, we summarize the present research situation home and abroad of the Diophantine equations x~3±27 =Dy~2. In the second chapter, we give the prior knowledge to the whole paper and give a brief introduction to the characters of Pell equations, the origin of recurrence sequence and the method of congruence. In the third chapter, we prove the whole integer solutions of Diophantine equations x~3±27 =Dy~2(D=7,13,19,26,31) in five sections. In section I, we prove that the Diophantine equation x~3 + 27 = 7y~2 has only integer solutions ( x ,y)=(-3,0), (1 ,±2 ) and the Diophantine equation x~3 - 27 =7y~2 has only integer solutions ( x ,y)=(3,0). In section II, we prove that the Diophantine equation x~3 + 27 = 13y~2 has only integer solution ( x ,y)=(-3,0), and the Diophantine equation x~3 - 27 = 13y~2 has only integer solution ( x ,y)=(3,0). In section III, we prove that the Diophantine equation x~3 + 27 = 19y~2 has only integer solutions ( x , y ) = ( -3 ,0), ( 24,±9), ( -2 ,±1) and the Diophantine equation x~3 - 27 = 19y~2 has only integer solution ( x ,y)=(3,0). In section IV, we prove that the Diophantine equation x~3 + 27 = 26y~2 has only integer solutions ( x , y ) = ( - 3,0),(1,±1 ),( - 719,±3781), and the Diophantine equation x~3 - 27 = 26y~2 has only integer solution ( x ,y)=(3,0). In section V, we prove that the Diophantine equation x~3 + 27 = 31y~2 has only integer solutions ( x ,y)=(-3,0) and the Diophantine equation x~3 - 27 = 31y~2 has only integer solution ( x ,y)=(3,0). In the fourth chapter, we summarize the total paper and put forward some problems which should be solved in the future.