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On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

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  • Pavel Trojovský

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

Abstract

Let F n be the n th Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k , such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p -adic valuation of Fibonacci numbers.

Suggested Citation

  • Pavel Trojovský, 2019. "On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence," Mathematics, MDPI, vol. 7(11), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:11:p:1073-:d:284746
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    References listed on IDEAS

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    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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    Cited by:

    1. Pavel Trojovský, 2021. "On Two Problems Related to Divisibility Properties of z ( n )," Mathematics, MDPI, vol. 9(24), pages 1-9, December.

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