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The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z ( n )

Author

Listed:
  • Eva Trojovská

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

  • Venkatachalam Kandasamy

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

Abstract

Let ( F n ) n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z ( n ) = min { k ≥ 1 : n ∣ F k } . Very recently, Trojovská and Venkatachalam proved that, for any k ≥ 1 , the number z ( n ) is divisible by 2 k , for almost all integers n ≥ 1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2 k by any integer m ≥ 1 . In this paper, in particular, we prove this conjecture.

Suggested Citation

  • Eva Trojovská & Venkatachalam Kandasamy, 2021. "The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z ( n )," Mathematics, MDPI, vol. 9(22), pages 1-7, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2912-:d:679917
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