Skip to content
DraconiticResearch
Search Sources API
194,226 studies31 sourcesUpdated just now
PreprintarXiv licencearXiv

The Fibonacci numbers are not an additive uniqueness set for multiplicative functions

  • Poo-Sung Park
Open at sourcePDFDOICite

Abstract

Let (Fn)n≥ 0 be the Fibonacci sequence. We show that a multiplicative function f satisfying f(Fn+Fm)=f(Fn)+f(Fm) (n,m≥ 1) need not be the identity function, even when f takes positive integer values. This answers negatively a question posed by Spiro in 1992. The smallest example presented here is obtained from F31=557· 2417. We prove the required divisibility equivalence, formulate an abstract prime-signature construction, and give a practical criterion producing further examples. We also describe the AI-assisted search that led to the construction and provide a reproducible certificate checker.

Related

  1. Diameter and radius of an additive-multiplicative graph
    Yi Hu, Nan Yang
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative α-stable processes: Existence, uniqueness and numerical approximation
    Jiaquan Lu, Huaizhong Zhao
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  3. An Integral Representation of the Generalized Fibonacci Numbers
    Bikash Chakraborty, Kalyan Barman, Soumon Roy +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  4. Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
    Marta Na Chen, Wenchang Chu
    AxiomsPeer-reviewedCC BY 4.0Algebra & number theoryCrossref
  5. The Fibonacci numbers are not 3-accessible
    William J. Wesley
    arXivPreprintCC BY-NC-ND 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{park2026fibonacci,
  title = {{The Fibonacci numbers are not an additive uniqueness set for multiplicative functions}},
  author = {Poo-Sung Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08137},
  eprint = {2609.08137},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08137},
}
RIS
TY  - UNPB
TI  - The Fibonacci numbers are not an additive uniqueness set for multiplicative functions
AU  - Poo-Sung Park
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08137
UR  - https://arxiv.org/abs/2609.08137
AB  - Let (F_n)_n≥ 0 be the Fibonacci sequence. We show that a multiplicative function f satisfying f(F_n+F_m)=f(F_n)+f(F_m) (n,m≥ 1) need not be the identity function, even when f takes positive integer values. This answers negatively a question posed by Spiro in 1992. The smallest example presented here is obtained from F_31=557· 2417. We prove the required divisibility equivalence, formulate an abstract prime-signature construction, and give a practical criterion producing further examples. We also describe the AI-assisted search that led to the construction and provide a reproducible certificate checker.
ER  -