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

Eva Trojovská and Venkatachalam Kandasamy
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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

Mathematics, 2021, vol. 9, issue 22, 1-7

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.

Keywords: order of appearance; fibonacci numbers; divisibility; natural density; prime numbers (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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