The Proof of a Conjecture Related to Divisibility Properties of z ( n )

Eva Trojovská and Kandasamy Venkatachalam
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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

Mathematics, 2021, vol. 9, issue 20, 1-8

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).

Keywords: order of appearance; fibonacci numbers; parity; 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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