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The Number of Terms in Greedy Fibonacci--Lucas Representations

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by Deep Bhattacharjee ↗, Ushashi Bhattacharya ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We prove Kimberling's conjecture on greedy representations by Fibonacci and Lucas numbers: the ratio of the number of Zeckendorf summands of 1,…,n1,\dots,n to the number of terms of their greedy representations tends to a limit in (1.2,1.4)(1.2,1.4), and the limit is (5−5)/2(5-\sqrt5)/2. We also prove the conjectures of Greathouse and Kurkov on the number of terms of the greedy representation.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Settles the Kimberling, Greathouse and Kurkov OEIS conjectures (A214981, A214973). The proofs are written out in the paper; the programs and their outputs are in the release. Both authors approved the final version. Code and data: release v2.0.0, https://github.com/creelie/musical-rotary-phone/releases/tag/v2.0.0.
AI provenance filesAI_PROVENANCE_KimberlingConjecture.md · 3 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

  1. Carlitz, L., Scoville, R., Hoggatt, V. E., Jr. (1972). Lucas representations. Fibonacci Quart. 10(1): 29--42. https://doi.org/10.1080/00150517.1972.12430968DOI
  2. de Moura, L., Ullrich, S. (2021). The Lean 4 theorem prover and programming language. In: Platzer, A., Sutcliffe, G., eds. Automated Deduction -- CADE 28 . Lecture Notes in Computer Science, Vol. 12699. Cham: Springer, pp. 625--635. https://doi.org/10.1007/978-3-030-79876-5_37DOI
  3. Kimberling, C. (2020). Lucas representations of positive integers. J. Integer Seq. 23: Article 20.9.5. https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.htmllink
  4. Koloğlu, M., Kopp, G. S., Miller, S. J., Wang, Y. (2011). On the number of summands in Zeckendorf decompositions. Fibonacci Quart. 49(2): 116--130. https://doi.org/10.1080/00150517.2011.12428056DOI
  5. Lekkerkerker, C. G. (1952). Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci. Simon Stevin 29: 190--195.
  6. The mathlib Community (2020). The Lean mathematical library. In: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs . New York: ACM, pp. 367--381. https://doi.org/10.1145/3372885.3373824DOI
  7. Pihko, J. (1988). On Fibonacci and Lucas representations and a theorem of Lekkerkerker. Fibonacci Quart. 26(3): 256--261. https://doi.org/10.1080/00150517.1988.12429631DOI
  8. OEIS Foundation Inc. (2026). The On-Line Encyclopedia of Integer Sequences. Published electronically at oeis.org (https://oeis.org).link

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026