Abstract
We consider applications of generalized golden sections to the geometric definition of a number and establish new properties of natural numbers that follow from this approach.
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REFERENCES
A. P. Stakhov, Introduction to Algorithmic Measurement Theory [in Russian], Sovetskoe Radio, Moscow (1977).
A. P. Stakhov, “The golden section in the measurement theory,” Comput. Math. Appl., 17, No.4–6, 613–638 (1989).
G. Bergman, “A number system with an irrational base,” Math. Mag., No. 31, 98–119 (1957).
A. P. Stakhov, “‘Golden’ section in digital technology,” Avtomat. Vychisl. Tekhn., No. 1, 27–33 (1980).
A. P. Stakhov, Codes of Golden Ratio [in Russian], Radio i Svyaz’, Moscow (1984).
N. N. Vorob’ev, Fibonacci Numbers [in Russian], Nauka, Moscow (1978).
E. V. Hoggat, Fibonacci and Lucas Numbers, Houghton-Mifflin, Palo Alto (1969).
A. P. Stakhov, V. Massingua, and A. A. Sluchenkova, Introduction to Fibonacci Coding and Cryptography, Osnova, Kharkov (1999).
V. M. Chernov and M. V. Pershina, “Fibonacci-Mersenne and Fibonacci-Fermat discrete transforms,” Bol. Inform. Golden Section: Theory Appl., No. 9–10, 25–31 (1999).
A. P. Stakhov and I. S. Tkachenko, “Fibonacci hyperbolic trigonometry,” Dokl. Nats. Akad. Nauk Ukr., Issue 7, 9–14 (1993).
A. P. Stakhov, “A generalization of the Fibonacci Q-matrix,” Dopov. Nats. Akad. Nauk Ukr., No. 9, 46–49 (1999).
A. P. Stakhov, “Brousentsov’s ternary principle, Bergman’s number system and ternary mirror-symmetrical arithmetic,” Comput. J., 45, No.2, 231–236 (2002).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1143–1150, August, 2004.
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Stakhov, A.P. Generalized golden sections and a new approach to the geometric definition of a number. Ukr Math J 56, 1362–1370 (2004). https://doi.org/10.1007/s11253-005-0064-3
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DOI: https://doi.org/10.1007/s11253-005-0064-3