
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, HR10002 Zagreb, Croatia 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)+1frac(x)) whose 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) 181198. ^{2} D. Svrtan, I. Urbiha, On explicit bijection between nonnegative integers and rationals and Knuth's problem, submitted. 