The Continued Fractions Ladder of Specific Pairs of Irrationals
Mitja Lakner
University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, 1000 Ljubljana, Slovenia.
Peter Petek
University of Ljubljana, Faculty of Education, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia.
Marjeta Škapin Rugelj *
University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, 1000 Ljubljana, Slovenia.
*Author to whom correspondence should be addressed.
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 
and 
with m a natural number. A big partial quotient in one sequence finds a connection in the other.
Keywords: Continued fraction, irrational number