Andrej Dujella: Newton's approximants and continued fractions

Let $d$ be a positive integer which is not a perfect square. Let $p_n/q_n$ denote the $n$-th convergent and $s(d)$ the length of the shortest period in the simple continued fraction expansion of $\sqrt{d}$. It was proved by Mikusinski in 1954 that if $s(d) \leq 2$, then all Newton's approximants

$$R_n = \frac{1}{2} \Big( \frac{p_n}{q_n} + \frac{dq_n}{p_n} \Big)$$

are convergents of $\sqrt{d}$, and moreover $R_n = p_{2n+1}/q_{2n+1}$ for all nonegative integers $n$. If $R_n$ is a convergent of $\sqrt{d}$, then we say that $R_n$ is a "good approximant". In 2001, we proved the converse of Mikusinski's result, namely that if all approximants are good, then $s(d) \leq 2$. It is easy to see that $R_n > \sqrt{d}$. Therefore, good approximants satisfy $R_n = p_{2n+1+2j}/q_{2n+1+2j}$ for an integer $j=j(d,n)$. If $s(d) \leq 2$, then $j(d,n) = 0$. For $s(d) > 2$, we proved the upper bound $\vert j(d,n)\vert \leq (s(d)-3)/2$, and we presented a sequence of $d$'s (given in terms of Fibonacci numbers) which shows that this upper bound for $\vert j(d,n)\vert$ is sharp. Let $b(d)$ denote the number of good approximants among the numbers $R_n$, $n = 0, 1, \ldots , s(d)-1$. We will present some results and conjectures (based on experimental data) about the magnitude of $b(d)$ compared with $d$ and $s(d)$.

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