Translator Disclaimer
June 2013 On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations
Tuangrat CHAICHANA, Takao KOMATSU, Vichian LAOHAKOSOL
Tokyo J. Math. 36(1): 239-251 (June 2013). DOI: 10.3836/tjm/1374497522

Abstract

Let $\xi=\sqrt{v/u}\tanh(uv)^{-1/2}$, where $u$ and $v$ are positive integers, and let $\eta=|h(\xi)|$, where $h(t)$ is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values $\eta$ of hyperbolic functions by rationals $x/y$ such that $x$ and $y$ satisfy Diophantine equations. We show that there are infinitely many coprime integers $x$ and $y$ such that $|y\eta-x|\ll\log\log y/\log y$ and a Diophantine equation holds simultaneously relating $x$ and $y$ and some integer $z$. Conversely, all positive integers $x$ and $y$ with $y\ge c_0$ solving the Diophantine equation satisfy $|y\eta-x|\gg\log\log y/\log y$.

Citation

Download Citation

Tuangrat CHAICHANA. Takao KOMATSU. Vichian LAOHAKOSOL. "On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations." Tokyo J. Math. 36 (1) 239 - 251, June 2013. https://doi.org/10.3836/tjm/1374497522

Information

Published: June 2013
First available in Project Euclid: 22 July 2013

zbMATH: 1283.11103
MathSciNet: MR3112386
Digital Object Identifier: 10.3836/tjm/1374497522

Subjects:
Primary: 11D09
Secondary: 11D25, 11J04, 11J70

Rights: Copyright © 2013 Publication Committee for the Tokyo Journal of Mathematics

JOURNAL ARTICLE
13 PAGES


SHARE
Vol.36 • No. 1 • June 2013
Back to Top