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

Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Mathematics 2020, 8(1), 124; https://doi.org/10.3390/math8010124
Received: 23 December 2019 / Revised: 10 January 2020 / Accepted: 13 January 2020 / Published: 14 January 2020
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. View Full-Text
Keywords: diophantine equation; asymptotic; Fibonacci numbers; order (rank) of appearance; p-adic valuation diophantine equation; asymptotic; Fibonacci numbers; order (rank) of appearance; p-adic valuation
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Trojovská, E. On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence. Mathematics 2020, 8, 124.

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