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The Number of Terms in Greedy Fibonacci--Lucas Representations

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

We prove Kimberling's conjecture on greedy representations by Fibonacci and Lucas numbers: the ratio of the number of Zeckendorf summands of $1,\dots,n$ to the number of terms of their greedy representations tends to a limit in $(1.2,1.4)$ , and the limit is $(5-\sqrt5)/2$ . We also prove the conjectures of Greathouse and Kurkov on the number of terms of the greedy representation.

A mix of human-written and AI-generated textHuman understanding: some partsFibonacci numbersLucas numbersZeckendorf representationformal verificationgreedy algorithmsummatory function

Oct 22, 2026PDF ↗View work →

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The Number of Terms in Greedy Fibonacci--Lucas Representations

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