Original scientific paper

Newton's approximants and continued fraction expansion of $\frac{1+\sqrt{d}}{2}$

Vinko Petričević ; Department of Mathematics, University of Zagreb, Zagreb, Croatia

Abstract

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 $R_n = \frac{p_n^2+\frac{d-1}4q_n^2}{q_n(2p_n-q_n)}$ 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 a 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.

Keywords

continued fractions; Newton's formula

Hrčak ID:

93236

URI

https://hrcak.srce.hr/93236

Publication date:

5.12.2012.

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