# Irrationality and transcendence of continued fractions with algebraic integers

## TL;DR

This paper generalizes previous results on the irrationality and transcendence of continued fractions with algebraic integer entries, establishing conditions under which such continued fractions are transcendental or irrational.

## Contribution

It extends earlier work by providing new growth conditions on algebraic integer sequences that guarantee the transcendence or irrationality of the resulting continued fractions.

## Key findings

- Certain growth conditions imply the continued fraction is transcendental.
- The results apply to sequences of algebraic integers with bounded degree.
- The work generalizes previous irrationality and transcendence criteria.

## Abstract

We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vert\alpha_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $\alpha = [0;\alpha_1, \alpha_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.

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Source: https://tomesphere.com/paper/1902.04312