Kyushu Journal of Mathematics
Online ISSN : 1883-2032
Print ISSN : 1340-6116
ISSN-L : 1340-6116
RATIONAL APPROXIMATIONS OF (1 + x)a AND APPLICATIONS TO THUE INEQUALITIES
Yuzen TANAKA
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2003 Volume 57 Issue 2 Pages 277-290

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Abstract

We study some Padé approximations to (1 + x)a and we give their lower bound using Rickert's theorem, where a and x are rational numbers in the open interval (0, 1) with (denominator of x) > (numerator of x)2×(denominator of a)3/2. Thenwe give an effective bound for the solutions of the Thue inequalities |Xn-(1+x)αY n| ≤ k for a positive integer n and rational numbers α, x with n ≥ 2 and 0 < α/n, x < 1 and a positive real number k. As an application we solve these inequalities for some special cases. Our result depends on the methods of Chudnovsky [C], Rickert [R] and Wakabayashi [W1, W2].

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© 2003 by Faculty of Mathematics, Kyushu University
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