An Unusual Continued Fraction

TL;DR

This paper studies a special real number defined by a unique continued fraction, proves its square is transcendental with a specific irrationality measure, and confirms a conjecture about the growth of its partial quotients.

Contribution

It introduces a novel continued fraction with a specific pattern, computes the irrationality measure of its square, and proves its transcendence along with growth properties of partial quotients.

Findings

The number's square has a finite irrationality measure.

Both the number and its square are transcendental.

Certain partial quotients grow doubly exponentially.

Abstract

We consider the real number $\sigma$ with continued fraction expansion $[a_0, a_1, a_2,\ldots] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots]$, where $a_i$ is the largest power of $2$ dividing $i+1$. We compute the irrationality measure of $\sigma^2$ and demonstrate that $\sigma^2$ (and $\sigma$) are both transcendental numbers. We also show that certain partial quotients of $\sigma^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.