4.7 Article

On the Parity of the Order of Appearance in the Fibonacci Sequence

期刊

MATHEMATICS
卷 9, 期 16, 页码 -

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MDPI
DOI: 10.3390/math9161928

关键词

order of appearance; Fibonacci numbers; parity; natural density; prime numbers

资金

  1. Excellence Project P.rF UHK, University of Hradec Kralove, Czech Republic [2213/2021-2022]

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The paper proves that the value of the order of appearance function z in the Fibonacci sequence is an even number for almost all positive integers.
Let (F-n)(n >= 0) be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z : Z(>= 1)-> Z(>= 1) is defined as z(n) := min{k >= 1 : F-k equivalent to 0 (mod n)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1).

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