实二次单位数方幂的整数部分

Integral Part of Powers of Real Quadratic Units

设D是非平方正整数,又设α=(u+v D)2,其中(u,v)是Pell方程u2-Dv2=1的正整数解.证明了:对于任何正整数n,[αn]都是奇数,[αn]+3都是平方数,其中[αn]是αn的整数部分.

Let D be a positive integer which is not a perfect square, and let a = （ u ＋ v √D）^2, where （ u, v ） is positive integer solution of the Pell equation u^2 - Dv^2 = 1. It is proved that, for any positive integer n, [ a^n ] is an odd integer and [ a^n ] ＋ 3 is a perfect square, where [ a^n ] is the integral part of a＇.