摘要
设α是正实数,[α_0;α_1,α_2。…]是α的简单连分数;d 是非平方整数,d_1、d_2是适合 d_1d_2=d,1≤d_1<d_2,gcd(d_1,d_2)=1的正整数,本文证明了:当且仅当α=(d_2/d_1)(1/2)时,α=[α_0α_1,…α_(n-1),2α_0],其中α_0>0,α_i=α_(n-1)(i=1…,α-1).
It is proved in this paper that α=[a<sub>0</sub>;a<sub>1</sub>,…,α<sub>n-1</sub>,2α<sub>0</sub>]if and only if α=（d<sub>2</sub>/d<sub>1</sub>）<sup>1/2</sup>,where a<sub>0</sub>>0,a<sub>i</sub>=a<sub>n-i</sub>,（i=1,…,n-1）,1<d<sub>1</sub><d<sub>2</sub>,gcd（d<sub>1</sub>,d<sub>2</sub>）=1 and the product d<sub>1</sub>d<sub>2</sub> is a positive integer which is not a quadrature.
