Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited

P. M. Voutier

doi:10.5802/jtnb.586

Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited

Number Theory 2012-02-01 v1 Classical Analysis and ODEs

Authors: P. M. Voutier

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Abstract

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$ , using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $\theta(k,l;x)$ and $\psi(k,l;x)$ for $k=1,3,4,6$ are also presented.

Keywords

hypergeometric functions function approximation rational functions

Cite

@article{arxiv.0802.1266,
  title  = {Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited},
  author = {P. M. Voutier},
  journal= {arXiv preprint arXiv:0802.1266},
  year   = {2012}
}

Comments

published version, but with some small changes, including typo in statement of Lemma 5.1(b), leading to simpler proof of Theorem 2.1

Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited

Abstract

Keywords

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