{"id":"hexagon/2610.00478","pdf":"https://hexagonmath.org/pdf/2610.00478v1","url":"https://hexagonmath.org/abs/2610.00478v1","tags":["msc:11B39","msc:11A67","msc:68V15","Fibonacci numbers","Lucas numbers","Zeckendorf representation","formal verification","greedy algorithm","summatory function"],"score":0,"title":"The Number of Terms in Greedy Fibonacci--Lucas Representations","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee","Ushashi Bhattacharya"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}