Rational Approximations to Certain Algebraic Numbers

Abstract

W.M.Schmit[11] conjectured that for any$\theta$ with deg$\theta \ge 3,$ there is no constant$C=C(\theta )$ so that$|p-q\theta |>C{q}^{-1}$ for every rationa$p/q.$ [12,p26] states that the computations of the first several thousand partial quotients for such numbers as$\sqrt[3]{2}$ and$\sqrt[3]{3}$ support the conjecture that the sequence of partial quotients is unbounded. In this paper, applying Dirichlet's approximation theorem to certain algebraic numbers$\theta ,$ e.g.$\theta =\sqrt[n]{d},d\in N,n\ge 3,d>0;$ ${\theta }^3+b_1\theta -b_0=0,b_0>0;$ ${\theta }^4+b_2{\theta }^2-b_0=0,b_0>0.$ We proved that there exists a effective constant$C=C(\theta )$ such that$|p-q\theta |>C{q}^{-1}$ for all$p/q.$ Our theorem shows their sequence of partial quotients can not be unbounded.