Abstract
Based on Zeckendorf’s theorem concerning the unique sum-representation of any positive integer in terms of Fibonacci numbers as well as Lucas numbers /1/, the purpose of this study is the development of relationships which enable prediction of the NUMBER of addends in these representations. Integral sequences {Fn/d} and {Ln/d} are considered such that d, with 2 is a predetermined integer and n is subject to appropriate conditions to assure integral elements in these sequences. Restrictions on n such that Fn = 0 (mod d) can always be determined. However, for n ε{5, 8, 10, 12, 13, 15, 16, 17, 20} there does not exist an n-value such that Ln = 0 (mod d).
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References
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© 1988 Springer Science+Business Media New York
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Freitag, H.T., Filipponi, P. (1988). On the Representation of Integral Sequences {Fn/d} and {Ln/d} as Sums of Fibonacci Numbers and as Sums of Lucas Numbers. In: Philippou, A.N., Horadam, A.F., Bergum, G.E. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7801-1_11
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DOI: https://doi.org/10.1007/978-94-015-7801-1_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8447-7
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