Glasnik matematički, Vol. 44 No. 2, 2009.
Izvorni znanstveni članak
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
Sažetak
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.
Ključne riječi
Diophantine approximation; Diophantine equations; trigonometric and hyperbolic functions
Hrčak ID:
44046
URI
Datum izdavanja:
9.12.2009.
Posjeta: 1.403 *