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The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic

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  • Stakhov, Alexey

Abstract

We consider two important generalizations of the golden proportion: golden p-proportions [Stakhov AP. Introduction into algorithmic measurement theory. Soviet Radio, Moscow, 1977 [in Russian]] and “metallic means” [Spinadel VW. La familia de números metálicos en Diseño. Primer Seminario Nacional de Gráfica Digital, Sesión de Morfología y Matemática, FADU, UBA, 11–13 Junio de 1997, vol. II, ISBN 950-25-0424-9 [in Spanish]; Spinadel VW. New smarandache sequences. In: Proceedings of the first international conference on smarandache type notions in number theory, 21–24 August 1997. Lupton: American Research Press; 1997, p. 81–116. ISBN 1-879585-58-8]. We develop a constructive approach to the theory of real numbers that is based on the number systems with irrational radices (Bergman’s number system and Stakhov’s codes of the golden p-proportions). It follows from this approach ternary mirror-symmetrical arithmetic that is the basis of new computer projects.

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  • Stakhov, Alexey, 2007. "The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 315-334.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:2:p:315-334
    DOI: 10.1016/j.chaos.2006.01.028
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    References listed on IDEAS

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    1. Stakhov, Alexey & Rozin, Boris, 2006. "The continuous functions for the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1014-1025.
    2. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    3. El Naschie, M.S., 2006. "Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 297-330.
    4. Stakhov, A.P., 2005. "The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 263-289.
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    Cited by:

    1. Chi Hongmei, 2013. "Generation of parallel modified Kronecker sequences," Monte Carlo Methods and Applications, De Gruyter, vol. 19(4), pages 261-271, December.
    2. Khan, Mohammad Nazrul Islam, 2021. "Novel theorems for the frame bundle endowed with metallic structures on an almost contact metric manifold," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    3. Cristina E. Hretcanu & Adara M. Blaga, 2021. "Types of Submanifolds in Metallic Riemannian Manifolds: A Short Survey," Mathematics, MDPI, vol. 9(19), pages 1-22, October.
    4. Falcon, Sergio & Plaza, Ángel, 2009. "k-Fibonacci sequences modulo m," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 497-504.
    5. Mohammad Nazrul Islam Khan & Uday Chand De & Teg Alam, 2023. "Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ -Manifolds," Mathematics, MDPI, vol. 11(14), pages 1-12, July.
    6. Basu, Manjusri & Prasad, Bandhu, 2009. "Coding theory on the m-extension of the Fibonacci p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2522-2530.
    7. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.
    8. Özkan, Engin & Kuloǧlu, Bahar & Peters, James F., 2021. "K-Narayana sequence self-Similarity. flip graph views of k-Narayana self-Similarity," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    9. Cristina E. Hretcanu & Mircea Crasmareanu, 2023. "The ( α , p )-Golden Metric Manifolds and Their Submanifolds," Mathematics, MDPI, vol. 11(14), pages 1-13, July.

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