On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Pavel Trojovský
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Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2019, vol. 7, issue 11, 1-10

Abstract: Let F n be the n th 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.

Keywords: diophantine equation; Fibonacci number; order of appearance; p-adic valuation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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