On the cubic Diophantine equation x~3+1=3py~2

Aim To study the positive integer solution of the Diophantine equation x3+1=3py2.Methods By using the basic properties of Pell equations.Results Let p be an odd prime with p≡1(mod 6).For p=3k2-2 or 3p=k2+2,where k is a positive integer,the equation x3+1=3py2 has no positive integer solution.Conclusion It is proved that the Diophantine equation has not integer solution for some special integers p.