Remarks and problems on finite and periodic continued fractions (Q1329054)

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.