On Two Problems Related to Divisibility Properties of z ( n )

Pavel Trojovský
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Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2021, vol. 9, issue 24, 1-9

Abstract: The order of appearance (in the Fibonacci sequence) function z : Z ≥ 1 → Z ≥ 1 is an arithmetic function defined for a positive integer n as z ( n ) = min { k ≥ 1 : F k ≡ 0 ( mod n ) } . A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z ( n ) = z ( n 2 ) , where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z ( n ) = z ( n + 1 ) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that lim sup n → ∞ ( z ( n + 1 ) − z ( n ) ) / ( log n ) 2 − ϵ = ∞ , for all ϵ ∈ ( 0 , 2 ) .

Keywords: order of appearance; Fibonacci numbers; divisibility; functional equation; prime numbers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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