The continued fractions ladder of specific pairs of irrationals

Abstract

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin's probability law. Results are given for sequences of partial quotients of positive irrational numbers $\xi$ and $m \over \xi$ with $m$ a natural number. A big partial quotient in one sequence finds a connection in the other.