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Newton's approximants and continued fraction expansion of (1+√d)/2

Let $d$ be a positive integer such that $d\equiv 1\pmod 4$ and $d$ is not a perfect square. It is well known that the continued fraction expansion of $\frac{; ; 1+\sqrt d}; ; 2$ is periodic and symmetric, and if it has the period length $\ell\le2$, then all Newton's approximants $R_n = \frac{; ; p_n^2+\frac{; ; d-1}; ; 4 q_n^2}; ; {; ; q_n(2p_n-q_n)}; ; $ are convergents of $\frac{; ; 1+\sqrt d}; ; 2$ and then it holds $R_n=\frac{; ; p_{; ; 2n+1}; ; }; ; {; ; q_{; ; 2n+1}; ; }; ; $ for all $n\ge0$. We say that $R_n$ is good approximant if $R_n$ is a convergent of $\frac{; ; 1+\sqrt d}; ; 2$. When $\ell>2$ then there is a good approximant in the half and at the end of the period. In this paper we prove that being a good approximant is a palindromic and a periodic property. We show that when $\ell>2$, there are $R_n$'s, which are not good approximants. Further, we define the numbers $j=j(d, n)$ by $R_n=\frac{; ; p_{; ; 2n+1+2j}; ; }; ; {; ; q_{; ; 2n+1+2j}; ; }; ; $ if $R_n$ is a good approximant and $b=b(d)=|{; ; n:0\le n\le \ell -1\text{; ; and $R_n$ is a good approximant}; ; }; ; |$. We construct sequences which show that $|j|$ and $b$ are unbounded.

Continued fractions ; Newton's formula.

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