On length of the period of the continued fraction of $n\sqrt{d}$

TL;DR

This paper investigates the behavior of the period length of continued fractions for multiples of quadratic irrationals, showing that the sequence of these lengths has infinitely many limit points as n varies.

Contribution

It proves that for non-square positive integers d, the sequence of period lengths of n√d's continued fractions has infinitely many limit points, revealing complex asymptotic behavior.

Findings

Sequence (D(n√d)) has infinitely many limit points.

Period length behavior is non-convergent and exhibits rich structure.

Results contribute to understanding of continued fraction expansions of quadratic irrationals.

Abstract

For a given quadratic irrational $\alpha$, let us denote by $D(\alpha)$ the length of the periodic part of the continued fraction expansion of $\alpha$. We prove that for a positive integer $d$, which is not a perfect square, the sequence $(D(n\sqrt{d}))_{n=1}^{\infty}$ has infinitely many limit points.