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The Proof of a Conjecture 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)

  • Kandasamy Venkatachalam

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

Abstract

The order of appearance of n (in the Fibonacci sequence) z ( n ) is defined as the smallest positive integer k for which n divides the k —the Fibonacci number F k . Very recently, Trojovský proved that z ( n ) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z ( n ) . In this paper, among other things, we prove that for any k ≥ 1 , the number z ( n ) is divisible by 2 k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Suggested Citation

  • Eva Trojovská & Kandasamy Venkatachalam, 2021. "The Proof of a Conjecture Related to Divisibility Properties of z ( n )," Mathematics, MDPI, vol. 9(20), pages 1-8, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2638-:d:659876
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