Abstract: MATH/CHEM/COMP 2002, Dubrovnik,
June 2429, 2002

On explicit bijection between nonnegative integers and rationals
AND Knuth's problem
Dragutin Svrtan Department
of Mathematics, Faculty of Science
and Mathematics, University of
Zagreb, Bijenička c. 30, HR10002 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 (without repetitions) all nonnegative rational numbers. A shorter recurrence is as follows: 1/q_{n} = _{└}q_{n}  1_{┘}+_{ }1 – {q_{n1}}. Then _{└}q_{n}  1_{┘}= ε_{2} (n) (=k if n is divisible by 2^{k} but not by 2^{k}^{+1}) leads to a solution of a recent problem of Knuth. A bijective discontinous function h: R_{0}^{+} → R^{+}, h(x) = 1/(2_{└}x_{┘} + 1 – x) is discovered, which realizes the sequence (q_{n})_{n}_{Î N}_{o}^{ }as one injective trajectory (through 0), since h^{om}(0) = q_{m}, m ³ 0. A wide Open Problem: Describe explicitly other trajectories of h in various number fields. In terms of the sawtooth function ((x)) one can write 1/h(x) = x – 2((x)) + d(x). The inverse of h is given by h^{1}(x) = 1/x – 2((1/x))  d(1/x). The results above and some conjectures (for quadratic extensions) are done jointly with I. Urbiha. 