On The Diophantine Equations X~3+M~3=DY~2

Diophantine equations is not only active in its own development, but also appliedto various areas of Discrete Mathematics widely.It plays a key role in studying andsolving actual problems.Therefore, many scholars carry out extensive and in-depthresearch on the Diophantine equations in the internal and the extenal.Many scholars have done a lot of work on the Diophantine equationsX~3+M~3=DY~2(M,D N),they focused on the integer solution of the Diophantineequations about D>0, d|D, d is not a square and d is not prime of theform6k+1,especially on M=1, D is prime of the form6k+1.However,because ofthe difficulty of solving such Diophantine equations,there is a vast study space on thisproblem.The main work of this paper is to discuss the integer solutions that theDiophantineequations x~3+M~3=Dy~2(M, D)=(-13,14),(3,61),(3, p),(±3,21),(-2,217),(2,111),(2,183)with congruent method, the Pell equations and recursivesequence and so on.In this paper we will prove that nontrivial integer solutions for the Diophantineequation x~3-133=14y~2has integer solutions (x, y)=(13,0),(117,±338); theDiophantine equation x~3+27=61y~2has integer solution (x,y)=(-3,0); theDiophantine equation x~3+27=py~2(p is the odd primes, p>0and p o1(mod12))has integer solution when3|x; the Diophantine equation x~3+27=21y~2has integersolutions (x,y)=(-3,0),(9,6), the Diophantine equation x~3-27=21y~2hasinteger solutions (x, y)=(3,0),(6,北3),(12,9),(66,117); the Diophantineequation x~3-8=217y~2has integer solutions (x, y)=(2,0),(50,24); the Diophantineequation x~3+8=111y~2has no integer solutions when (x, y)=1. the Diophantineequation x~3+8=183y~2has no integer solutions when (x, y)=1.