Rational approximations to 2 3 and other algebraic numbers revisited

Paul M. Voutier[1]

  • [1] London, UK

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 263-288
  • ISSN: 1246-7405

Abstract

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In this paper, we establish improved effective irrationality measures for certain numbers of the form n 3 , 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 the Chebyshev functions in arithmetic progressions θ ( k , l ; x ) and ψ ( k , l ; x ) for k = 1 , 3 , 4 , 6 are also presented.

How to cite

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Voutier, Paul M.. "Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 263-288. <http://eudml.org/doc/249947>.

@article{Voutier2007,
abstract = {In this paper, we establish improved effective irrationality measures for certain numbers of the form $\@root 3 \of \{n\}$, 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 the Chebyshev functions in arithmetic progressions $\theta (k,l;x)$ and $\psi (k,l;x)$ for $k=1,3,4,6$ are also presented.},
affiliation = {London, UK},
author = {Voutier, Paul M.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {irrationality measure; algebraic number},
language = {eng},
number = {1},
pages = {263-288},
publisher = {Université Bordeaux 1},
title = {Rational approximations to $\{\@root 3 \of \{2\}\}$ and other algebraic numbers revisited},
url = {http://eudml.org/doc/249947},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Voutier, Paul M.
TI - Rational approximations to ${\@root 3 \of {2}}$ and other algebraic numbers revisited
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 263
EP - 288
AB - In this paper, we establish improved effective irrationality measures for certain numbers of the form $\@root 3 \of {n}$, 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 the Chebyshev functions in arithmetic progressions $\theta (k,l;x)$ and $\psi (k,l;x)$ for $k=1,3,4,6$ are also presented.
LA - eng
KW - irrationality measure; algebraic number
UR - http://eudml.org/doc/249947
ER -

References

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  11. G. Lettl, A. Pethő, P. M. Voutier, Simple Familiies of Thue Inequalities. Trans. Amer. Math. Soc. 351 (1999), 1871–1894. Zbl0920.11041MR1487624
  12. E. M. Nikishin, Arithmetic properties of the Markov function for the Jacobi weight. Anal. Math. 8 (1982), 39–46. Zbl0495.10022MR662702
  13. O. Ramaré, R. Rumely, Primes in arithmetic progressions. Math. Comp. 65 (1996), 397–425. Zbl0856.11042MR1320898
  14. M. Rubinstein, http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/ZEROS 
  15. A. Togbe, P. M. Voutier, P. G. Walsh, Solving a family of Thue equations with an application to the equation x 2 - D y 4 = 1 . Acta. Arith. 120 (2005), 39–58. Zbl1155.11318MR2189717

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