On the Diophantine Equation z ( n ) = (2 ? 1/ k ) n Involving the Order of Appearance in the Fibonacci Sequence

Eva Trojovská
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Eva Trojovská: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2020, vol. 8, issue 1, 1-8

Abstract: Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n . In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k .

Keywords: diophantine equation; asymptotic; Fibonacci numbers; order (rank) of appearance; p-adic valuation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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