The Diophantine equation x~3-5~3=3py~2

Let p be a fixed prime.By using some elementary number theory methods,it is proved that the equation x3-53=3py2 has positive integer solution(x,y)with gcd(x,y)=1if and only if p=Q(27a4+45a2+25),where ais a positive integer and Q(27a4+45a2+25)is the square free factor of 27a4+45a2+25.Thus,if p≠7or 13(mod30),the equation has no positive integer solutions(x,y)with gcd(x,y)=1.