Original scientific paper

https://doi.org/10.3336/gm.44.2.02

On convergents formed from Diophantine equations

Carsten Elsner ; FHDW, Fachhochschule für die Wirtschaft, University of Applied Sciences, Freundallee 15, D-30173 Hannover, Germany
Takao Komatsu ; Graduate School of Science and Technology, Hirosaki University, Hirosaki, 036-8561 Japan
Iekata Shiokawa ; Department of Mathematics, Keio University, Yokohama, 223-8522 Japan

Abstract

We compute upper and lower bounds for the approximation of certain values ξ of hyperbolic and trigonometric functions by rationals x/y such that x, y satisfy Diophantine equations. We show that there are infinitely many coprime integers x, y such that

|y ξ - x| (log log y)/(log y)

and a Diophantine equation holds simultaneously relating x, y and some integer z. Conversely, all positive integers x, y with y ≥ c0 solving the Diophantine equation satisfy
|y ξ - x| (log log y)/(log y)

Moreover, we approximate sin(πα) and cos(πα) by rationals in connection with solutions of a quadratic Diophantine equation when tan(πα/2) is a Liouville number.

Keywords

Diophantine approximation; Diophantine equations; trigonometric and hyperbolic functions

Hrčak ID:

44046

URI

https://hrcak.srce.hr/44046

Publication date:

9.12.2009.

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