On The Diophantine Equations X³±P^3k=DY²

In this paper the author illustrates from four chapters:The first chapter reviews research status of the Diophantine equationX³±P^3k=DY²(D>0).The second chapter gives the preparation knowledge of the paper, and introducesthe nature of the Pell equation, recursive sequence and simple congruence method.The third chapter is mainly divided into four parts for solution of DiophantineequationX³±P^3k=DY²while p, D is given different integer.On theDiophantine equationnX³±23^3k=DY²(Where D>0, d|D, d is not a square and d isnot prime of the form6k+1),the author has got an recursive formula about k in thefirst section. By using the method of recurrent sequence and guadratic remaider theDiophantine equationsx323313y2have only integer solutions (x, y)(23,0)inthe second section.The Diophantine equationsX³±8=73Y²is proven to have nointeger solution with (x, y)=1in the third section,so D=67has not been improvedwith D <80. By using the method of recurrent sequence and proven conclusion,theequationX³±27=43Y²has only integer solutions (x, y)(3,0),(240,567)in thefourth section.The fourth chapter has made the summary and put forward to possible futuredevelopment about the issue.