关于Pell数列的Walsh猜想

On Walsh conjecture concerning Pell sequences

设D是非平方正整数,u1,+v1 D是Pell方程u2-Dv2=1的基本解.对于正整数n,设un,vn是适合un+vn D=(u1+v1 D)n的正整数.证明了当D≠22r·1785,其中r∈{0,1,2},而v1是奇数时,如果vn=2z2,其中z是正整数,则n=2.

Let D be a nonsquare positive integer, and let u1 + v1D be the fundamental solution of the Pell equation u2- Dv2=1 .For any positive integer n,let un and vn be positive integers such that un+vn D=(u1+v1 D )n . In this paper we prove that in D=22r.1785, where r |0,1,2|,v1 is odd and vn =2z2, where n ,z are positive integers, then n=2.