On the rational approximations to the powers of an algebraic number

Details On the rational approximations to the powers of an algebraic number On the rational approximations to the powers of an algebraic number

2004-03-30

arxiv.org/abs/math/0403522v1

Mathematics

Number Theory

12 pages, plain Tex

Scientific paper

About fifty years ago Mahler proved that if $\alpha>1$ is rational but not an integer and if $0l^n$ apart from a finite set of integers $n$ depending on $\alpha$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $\alpha$ such that the continued fraction of $\alpha^n$ has period length tending to infinity.

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