Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002



Recounting rationals, Knuth's problem and some conjectures


Dragutin Svrtan and Igor Urbiha


Department of  Mathematics, Faculty of Science and  Mathematics, University of Zagreb, Bijenička c. 30, HR-10002 Zagreb, Croatia

We show that a sequence defined by q_0=0, q_2m= q_m/(1+q_m) (m>0), and q_(2m+1)=1+q_m generates all nonnegative rational numbers (without repetitions).

More generally we discover a ('new') bijective discontinuous function h from nonnegative reals to positive ones which is defined by h(x)=1/(floor(x)+1-frac(x)) whose
forward orbit of 0 realizes the above sequence (q_n).

We pose a (wide) Open Problem: Describe explicitly other orbits of h in various number field extensions of the field of rational numbers.

For quadratic extensions we state some results (including a solution of a recent problem of Knuth) and some intriguing conjectures.


1 I. Urbiha, Some properties of a function studied by De Rham, Carlitz and Dijkstra and its relation to the (Eisenstein-)Stern's diatomic sequence, Mathematical Communications 6 (2001) 181-198.


2 D. Svrtan, I. Urbiha, On explicit bijection between nonnegative integers and rationals and Knuth's problem, submitted.