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On the Diophantine Equation z ( n ) = (2 − 1/ k ) n Involving the Order of Appearance in the Fibonacci Sequence

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  • Eva Trojovská

    (Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic)

Abstract

Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n . In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k .

Suggested Citation

  • Eva Trojovská, 2020. "On the Diophantine Equation z ( n ) = (2 − 1/ k ) n Involving the Order of Appearance in the Fibonacci Sequence," Mathematics, MDPI, vol. 8(1), pages 1-8, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:124-:d:308482
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    References listed on IDEAS

    as
    1. Diego Marques, 2011. "On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-4, April.
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