Continued fractions and transcendental numbers
Number Theory
2012-05-07 v1
Abstract
It is widely believed that the continued fraction expansion of every irrational algebraic number either is eventually periodic (and we know that this is the case if and only if is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are derived thanks to new combinatorial transcendence criteria recently obtained by Adamczewski and Bugeaud.
Cite
@article{arxiv.math/0511682,
title = {Continued fractions and transcendental numbers},
author = {Boris Adamczewski and Yann Bugeaud and Les J. L. Davison},
journal= {arXiv preprint arXiv:math/0511682},
year = {2012}
}