Far Eastern Mathematical Journal

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On two relations characterizing the golden ratio

Zhukova A.A., Shutov A.V.

2021, issue 2, P. 194–202 DOI: https://doi.org/10.47910/FEMJ202116

Abstract

V.G. Zhuravlev found two relations associated with the golden ratio: $\tau=\frac{1+\sqrt{5}}{2}$: $[([i\tau]+1)\tau]=[i\tau^2]+1$ and $[[i\tau]\tau]+1=[i\tau^2]$. We give a new elementary proof of these relations and show that they give a characterization of the golden ratio. Further we consider satisfability of our relations for finite sets of $i$-s and establish some forcing property for this situation.

Keywords: golden ratio, Fibonacci numbers

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References

[1] V.G. Zhuravlev, “Odnomernye razbieniia Fibonachchi”, Izvestiia RAN. Seriia matematicheskaia, 71:2 (2007), 89–122. [2] A.V. Shutov, “Perenormirovki vrashchenii okruzhnosti”, Chebyshevskii sbornik, 5:4 (2004), 125–143. [3] R. Grekhem, D. Knut, O. Patashnik, Konkretnaia matematika. Osnovanie informatiki, BINOM. Laboratoriia znanii, M., 2009. [4] E. Zeckendorf, “Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas”, Bulletin de la Societe Royale des de Liege, 41 (1972).

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