Repdigits as Product of Terms of k -Bonacci Sequences

Petr Coufal and Pavel Trojovský
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Petr Coufal: Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic
Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic

Mathematics, 2021, vol. 9, issue 6, 1-10

Abstract: For any integer k ? 2 , the sequence of the k -generalized Fibonacci numbers (or k -bonacci numbers) is defined by the k initial values F ? ( k ? 2 ) ( k ) = ? = F 0 ( k ) = 0 and F 1 ( k ) = 1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form a a … a , with a ? [ 1 , 9 ] ) in the sequence ( F n ( k ) F n ( k + m ) ) n , for m ? [ 1 , 9 ] . This result generalizes a recent work of Bedna?ík and Trojovská (the case in which ( k , m ) = ( 2 , 1 ) ). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method).

Keywords: k -generalized Fibonacci numbers; linear forms in logarithms; reduction method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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