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arXiv · 1505.00667

An Unusual Continued Fraction

Dzmitry Badziahin·Jeffrey Shallit
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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.

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Dzmitry Badziahin, Jeffrey Shallit. 2015-05-04. An Unusual Continued Fraction. https://arxiv.org/abs/1505.00667

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