engleski

Recounting Rationals, Knuth's Problem and some Conjectures

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 non-negative rational numbers. A shorter recurrence is as follows: 1/q_n =floor(q_(n - 1))+1– frac(q_(n-1)). Then floor(q_(n - 1))= e_2(n) (=k if n is divisible by 2^k but not by 2^(k+1) leads to a solution of a recent AMM problem 10006 of Knuth. A bijective discontinous function h: R_0^(+) → R^(+), h(x) = 1/(2floor(x)+1-x) is discovered, which realizes the sequence (q_n) as one injective trajectory (through 0), since (m-th iterate of h)(0) = q_m, m >= 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 are also considered.

bijection; rationals; sawtooth function; Knuth's problem

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