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On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Mathematics 2019, 7(11), 1073; https://doi.org/10.3390/math7111073
Received: 28 September 2019 / Revised: 3 November 2019 / Accepted: 6 November 2019 / Published: 7 November 2019
(This article belongs to the Section Mathematics and Computers Science)
Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + for | | { 1 , , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers. View Full-Text
Keywords: diophantine equation; Fibonacci number; order of appearance; p-adic valuation diophantine equation; Fibonacci number; order of appearance; p-adic valuation
MDPI and ACS Style

Trojovský, P. On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence. Mathematics 2019, 7, 1073.

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