Abstract. It has been believed that the continued fraction expansion of (α, β) (1, α, β is a Q-basis of a real cubic field) obtained by the modified JacobiPerron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of () ( x denoting the fractional part of x). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of (α, β) with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields.
IntroductionThe study of continued fractions has a long history dating back to J. Wallis (1616-1703) and Ch. Huygens (1629-1695) [8]. In particular, many kinds of higherdimensional continued fractions have been studied starting with K.G. Jacobi (1804-1851) [7]. A central problem has been to find a higher-dimensional generalization of Legendre's theorem concerning the periodic continued fractions. In fact, the following conjecture has been believed. 1, α 1 Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of (
Conjecture. Let) by the modified Jacobi-Perron algorithm.In this paper, we give some candidates of algorithms of continued fraction expansion of dimensions 2 and 3, which can be easily generalized to any dimension.By numerical experiments, we checked that, for instance, ( √ m / ∈ Q) obtained by our algorithm become periodic. We showed the periodicity
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