关于丢番图方程x^3-y^6=3pz^2的整数解

On the Positive Integer Solutions of the Diophantine Equation x^3-y^6=3pz^2

设p≡5(mod 6)为素数.证明了丢番图方程x3-y6=3pz2在p≡5(mod 12)为素数时均无正整数解;在p≡11(mod 12)为素数时均有无穷多组正整数解,并且还获得了该方程全部正整数解的通解公式,同时还给出了该方程的部分整数解.

In this paper, we study the integer solutions of the diophantine equation χ^3一У^6=3pz^2, where p=5 （ rood 6） is some prime number. We prove that the equation has no positive integer solution if p=5（mod 12）, and the equation has infinitely many positive integer solutions if p=11 （mod 12）. For the second case, the general formula of the positive integer solutions and parts of the solutions are given.