Mendès France, Michel Remarks and problems on finite and periodic continued fractions. (English) Zbl 0808.11007 Enseign. Math., II. Sér. 39, No. 3-4, 249-257 (1993). The author raises eight interesting problems with varying difficulty. The first problem concerns properties of Pisot numbers. Two further problems are as follows:(1) Let \(\delta(x)\) be the length of the continued fraction expansion (c.f.e.) of the rational \(x\). Is it true that for all integers \(a\) and \(b\), \(1<a<b\), \((a,b)=1\), we have \[ \lim_{n\to\infty} 1/n\;\delta ((a/b)^ n)= 12/\pi^ 2 \ln 2\ln b\;? \] (2) Let \(x\) be a real quadratic number and let \(\pi(x)\) be the period of the c.f.e. of \(x\). Is it true that \(\sup_ n \pi(x^ n)= \infty\)?This last problem has now been partially solved. Reviewer: T.Tonkov (Sofia) Cited in 7 Documents MSC: 11A55 Continued fractions Keywords:Möbius maps; Pisot numbers; length; continued fraction expansion; period PDFBibTeX XMLCite \textit{M. Mendès France}, Enseign. Math. (2) 39, No. 3--4, 249--257 (1993; Zbl 0808.11007)