Extremal orders of the Zeckendorf sum of digits of powers

Details Extremal orders of the Zeckendorf sum of digits of powers Extremal orders of the Zeckendorf sum of digits of powers

2010-09-27

arxiv.org/abs/1009.5360v1

Mathematics

Number Theory

11 pages

Scientific paper

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>= 2. We use this to show that for all $>N_0(h) there is an n such that n is the sum of N Fibonacci numbers and n^h is the sum of at most 130 h^2 Fibonacci numbers. Moreover, we give upper and lower bounds on the number of n's with small and large values of s_F(n^h)/s_F(n). This extends a problem of Stolarsky to the Zeckendorf representation of powers, and it is in line with the classical investigation of finding perfect powers among the Fibonacci numbers and their finite sums.

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