We compute upper and lower bounds for the approximation of hyperbolic functions at points by rationals , such that satisfy a quadratic equation. For instance, all positive integers with solving the Pythagorean equation satisfy
Conversely, for every there are infinitely many coprime integers , such that
and hold simultaneously for some integer . A generalization to the approximation of for rational functions is included.
Nous calculons des bornes supérieures et inférieures pour l’approximation de fonctions hyperboliques aux points par des rationnels , tels que satisfassent une équation quadratique. Par exemple, tous les entiers positifs avec , solutions de l’équation de Pythagore , satisfont
Réciproquement, pour chaque , il existe une infinité d’entiers , premiers entre eux, tels que
et soient réalisés simultanément avec entier. Une généralisation à l’approximation de , pour fonction rationnelle, est incluse.
@article{JTNB_2007__19_2_393_0, author = {Elsner, Carsten and Komatsu, Takao and Shiokawa, Iekata}, title = {Approximation of values of hypergeometric functions by restricted rationals}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {393--404}, publisher = {Universit\'e Bordeaux 1}, volume = {19}, number = {2}, year = {2007}, doi = {10.5802/jtnb.593}, zbl = {1167.11026}, mrnumber = {2394893}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.593/} }
TY - JOUR AU - Elsner, Carsten AU - Komatsu, Takao AU - Shiokawa, Iekata TI - Approximation of values of hypergeometric functions by restricted rationals JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 393 EP - 404 VL - 19 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.593/ DO - 10.5802/jtnb.593 LA - en ID - JTNB_2007__19_2_393_0 ER -
%0 Journal Article %A Elsner, Carsten %A Komatsu, Takao %A Shiokawa, Iekata %T Approximation of values of hypergeometric functions by restricted rationals %J Journal de théorie des nombres de Bordeaux %D 2007 %P 393-404 %V 19 %N 2 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.593/ %R 10.5802/jtnb.593 %G en %F JTNB_2007__19_2_393_0
Elsner, Carsten; Komatsu, Takao; Shiokawa, Iekata. Approximation of values of hypergeometric functions by restricted rationals. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 393-404. doi : 10.5802/jtnb.593. http://archive.numdam.org/articles/10.5802/jtnb.593/
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