关于丢番图方程x^3±3~3=3pqy^2

On the Diophantine Equation x^3± 3~3=3pqy^2

主要利用递归数列、同余式、平方剩余以及Pell方程解的性质证明了当p≡1(mod 12),q≡12s2+1时,丢番图方程x3-33=3pqy2仅有整数解(x,y)=(3,0);当p≡1(mod 24),q≡12s2+1时,丢番图方程x3+33=3pqy2仅有整数解(x,y)=(-3,0).

In this paper,the authors mainly use recurrent sequence,congruence sequece,quadratic residue andsome properties of the solutions to Pell equation to prove that when p≡1（mod 12）,q≡12s2＋1,the Diophantineequation x3-33=3pqy2 only has integer solution（x,y）=（3,0）;when p≡1（mod 24）,q≡12s2＋1,the Diophantineequation x3＋33=3pqy2 only has integer solution（x,y）=（-3,0）.