On The Diophantine Equations X³±P^3k=Dy²

In thi s paper the author illustrates from four chapters:The first chapter reviews research status of the Diophantine equation χ3±Ρ3=Dy2(D>0).The second chapter gives the preparation knowledge of the paper,and introduces the nature of the Pell equation,recursive sequence and simple congruence method.The third chapter is mainly diVided into four part for solution of Diophantine equation χ3±Ρ3=DY2while P,D is given different integer.On the Diophantine equations χ3±113k=Dy2(Where D>0,d|D,d is not a square and dis not prime of the form6K+1,K≥2):The author has got an. On the Diophantine equations,χ3±113=Dy2(Where D>0,d|D,d is not a square and dis not prime of the form6K+1)in the first section:Then using the recursive formula about K,the author has got the method of their nontrivial integer solutions out.In the second section by using the method of recurrent sequence and guadratic remaider,the Diophantine equations,χ3-113=7y2is proven to have integer solution (x,y)=(-11,0):the Diophantine equation x3-8=91y2has only integer solutions (x,y)=(1,0)and(9,土2);the Diophantine equation χ2+27=67y2as only integer solutions(x,y)=(-3,0),(1320,土5859).The f.ourth chapter has made the summary and put forward to possible future development about the issue.