Tokyo Journal of Mathematics

On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations


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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 $|y\eta-x|\gg\log\log y/\log y$.

Article information

Tokyo J. Math., Volume 36, Number 1 (2013), 239-251.

First available in Project Euclid: 22 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations
Secondary: 11D25: Cubic and quartic equations 11J04: Homogeneous approximation to one number 11J70: Continued fractions and generalizations [See also 11A55, 11K50]


CHAICHANA, Tuangrat; KOMATSU, Takao; LAOHAKOSOL, Vichian. On Convergents of Certain Values of Hyperbolic Functions Formed from Diophantine Equations. Tokyo J. Math. 36 (2013), no. 1, 239--251. doi:10.3836/tjm/1374497522.

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