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

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

2008-02-09

arxiv.org/abs/0802.1266v1

Journal de Th\'eorie des Nombres de Bordeaux 19 (2007), 265-288

Mathematics

Number Theory

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

Scientific paper

10.5802/jtnb.586

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.

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