On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

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 8, 1-10

Abstract: The k -generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k -bonacci or k -step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Keywords: k -generalized Fibonacci sequence; Diophantine equation; linear form in logarithms; continued fraction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (12)

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