Geodesic
On two relations characterizing the golden ratio
Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 2, pp. 194-202.

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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. 
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A. A. Zhukova; A. V. Shutov. On two relations characterizing the golden ratio. Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 2, pp. 194-202. http://geodesic.mathdoc.fr/item/DVMG_2021_21_2_a5/

[1] V. G. Zhuravlev, “Odnomernye razbieniya Fibonachchi”, Izvestiya RAN. Seriya matematicheskaya, 71:2 (2007), 89–122 | MR | Zbl

[2] A. V. Shutov, “Perenormirovki vraschenii okruzhnosti”, Chebyshevskii sbornik, 5:4 (2004), 125–143 | MR | Zbl

[3] R. Grekhem, D. Knut, O. Patashnik, Konkretnaya matematika. Osnovanie informatiki, BINOM. Laboratoriya znanii, M., 2009

[4] E. Zeckendorf, “Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas”, Bulletin de la Société Royale des de Liège, 41 (1972) | MR