关于Pell数的Diophantine方程x^2+U_n^y=V_n^z

The Diophantine Equation x^2+U_n^y=V_n^z on Pell Numbers

设n是正奇数,Un=(αn+βn)/2,Vn=(αn-βn)/2 2～(1/2),其中α=1+2,β=1-2.运用Pell数的算术性质讨论了方程x2+Uyn=Vzn的正整数解(x,y,z),证明了当n≡±3(mod 8)时,该方程仅有正整数解(x,y,z)=(V2n-1,2,4).

Let n be a positive odd integer,and let Un=（αn＋β n）/2,Vn=（αn-βn）/2 2～（1/2）,where α=1＋2～（1/2）,β=1-2～（1/2）.In this paper,using the arithmetic properties of Pell numbers,the positive integer solutions（x,y,z） of the equation x2＋Uyn=Vzn are discussed,and prove that if n≡±3（mod 8）,then the equation has only the positive integer solution（x,y,z）=（Vn2-1,2,4）.