The rational part of a periodic continued fraction

TL;DR

This paper investigates the relationship between the digits of a periodic continued fraction and the properties of its quadratic irrational value, revealing a special link between the last digit of the repeating block and the rational part of the number.

Contribution

It identifies a specific connection between the last digit of the repeating block and the rational part of the quadratic irrational in periodic continued fractions, especially when twice the rational part is an integer.

Findings

The magnitude of 2a is determined by the last digit c_n.

The fractional part of 2a is independent of c_n.

2a is an integer if and only if the sequence c_1,...,c_{n-1} is symmetric.

Abstract

Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The numbers $a$ and $\sqrt b$ are uniquely determined by $x$. In general it is difficult to say what the influence of a certain digit of the repeating block on the appearance of $x$ is. We highlight a noteworthy exception from this rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit $c_n$ of the repeating block, the fractional part of $2a$, however, is independent of $c_n$. Of particular interest is the case $2a\in\mathbb Z$, which occurs if, and only if, the sequence $c_1,\ldots,c_{n-1}$ is symmetric.